Sharedwww / breuil / report-03-28-01.texOpen in CoCalc
Author: William A. Stein

\documentclass{article}
\usepackage{amsmath}
\usepackage{fullpage}
\title{Some Data for C. Breuil About the Local Representation Attached to a Newform of Medium Weight}
\author{David Savitt \and William A. Stein}
\newcommand{\C}{\mathbf{C}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\Qbar}{\overline{\mathbf{Q}}}
\newcommand{\Qellbar}{\overline{\mathbf{Q}}_\ell}
\newcommand{\Frob}{{\rm Frob}}
\newcommand{\trace}{{\rm trace}}
\newcommand{\SL}{{\rm SL}}
\begin{document}
\maketitle

\section{Introduction}

This document contains data about $\rho_{f}|D_\ell$, where~$f$ is
a mod-$\ell$ newform whose weight~$k$ satisfies $\ell+1 < k \leq 2\ell$.
The data is organized as follows.
Section~$\ell$, for $\ell=2,3,5,7,11,13,17$
contains examples in characteristioc $\ell$.
Each section is divided up into subsections corresponding to levels~$N$.
In each subsection, there is a bullet corresponding to each mod-$\ell$
newform~$f$ whose weight~$k$ satisfies $\ell+1 < k \leq 2\ell$.
If~$f$ is ordinary or the representation attached to~$f$ is reducible,
then we do no further analysis.  If~$f$ is supersingular,
we search for a cuspidal newform~$g$ of weight $k'$, where $2 \leq k' \leq \ell+1$, such that $\rho_f$ is a twist of $\rho_g$ by a power of the mod-$\ell$
cyclotomic character. In fact, we really only check that
$\trace((\rho_f\otimes \chi^i)(\Frob_p)) = \trace(\rho_g(\Frob_p))$
for all primes $p\neq \ell$ that are
less than $[\SL_2(\Z):\Gamma_0(N)]\cdot (\ell/6)+10$.
If no such~$g$ exists, then (I think!) this implies that $\rho_f$ is
reducible (or old?), so we do not consider it further.  When such a~$g$ exists,
we apply results of Deligne and Fontaine to read off the semisimplification of $\rho_g|D_\ell$ from
the $q$-expansion and weight of~$g$.  By twisting, we obtain the semisimplification
of $\rho_f|D_\ell$.

\begin{center}
{\bf WARNING:} This document is very long.
\end{center}

Many thanks. I [Breuil] also have 2 questions:
\begin{enumerate}
\item {\em Assuming~$f$ is a new form on $\Gamma_0(N)$ (thus eigenvector for all
$T_p$ and $U_p$), I guess so is~$g$?}\\
The program computes the eigenforms~$f$ and~$g$ as follows.  First it lists
the newforms (for all $T_p$ and $U_p$) in $S_2(\Gamma_0(N);\C)$.  Then it
computes all of the reductions of these newforms to mod-$\ell$ eigenforms.
Thus both~$f$ and~$g$ are attached to new maximal ideals of residue characteristic~$\ell$
of the full Hecke algebra associated to $S_2(\Gamma_0(N);\C)$.

\item {\em Is the lift of~$f$ you give a classical eigenform or a genuine $\ell$-adic one?
(I want to be sure that when the slope of~$f$ is~$1$, $(a_\ell/\ell) \mod{\ell}$
computed on the $\ell$-adic lift you give is equal to
$(a_\ell/\ell) \mod{\ell}$ computed on a classical'' lift.
It seems to be so in several cases I have checked).}\\
I do not completely understand the question.  However, I can tell you how
the lift~$f_0$ of~$f$ is computed by my program.  The program computed~$f$
by finding the newforms in $S_2(\Gamma_0(N);\C)$, so it knows a
newform $\tilde{f} \in S_2(\Gamma_0(N);\Qbar)$ that has~$f$ as one of
its reductions.  To find $f_0$, the program first computes each embeddings of $\tilde{f}$
into $\Qellbar[[q]]$.  It then compares the mod-$\ell$ reduction
of each of these $\ell$-adic modular forms with~$f$, and lets $f_0$ be
the first form it finds with mod-$\ell$ reduction equal to~$f$.

\end{enumerate}

\subsection{Technical remarks}
This data was created using the MAGMA program that can be found at the end of this document.
It was necessary to write a number of routines to convert the output of MAGMA into something that tex
understands.  This got really tedious, so I haven't aimed for perfection.  {\bf For example,
sometimes you'll see things like $O(q^205)$, which should really be $O(q^{205})$!}
I hope this doesn't cause undue confusion.

I have no reason to suspect that this data is incorrect in any way.  However, it relies on
several layers of very complicated computer programs, so be very careful!!

\tableofcontents

\setcounter{section}{1}
\section{$\mathbf{2 \leq N \leq 50, \quad \ell = 2}$}

\subsection{$\mathbf{N = 3, \quad \ell = 2}$ \quad (no forms)}

There are no forms to analyze.

\subsection{$\mathbf{N = 5, \quad \ell = 2}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(5);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 7, \quad \ell = 2}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(7);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 9, \quad \ell = 2}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(9);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 11, \quad \ell = 2}$ \quad (1 form)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(11);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -2^2*a + 3q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -2x + -2$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 13, \quad \ell = 2}$ \quad (3 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(13);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(13);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(13);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 15, \quad \ell = 2}$ \quad (2 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(15);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(15);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 17, \quad \ell = 2}$ \quad (3 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(17);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(21839*2^-2 + O(2^18))*a - 60961*2 + O(2^18)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -2729*2^3 + O(2^20)x + -15241*2^5 + O(2^20)$.
The slope of $f_0$ is $5/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(17);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(17);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 19, \quad \ell = 2}$ \quad (3 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(19);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + (38281*2^2 + O(2^20))*a - 54941*2 + O(2^20)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 28711*2^4 + O(2^20)x + -164833*2 + O(2^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(19);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(19);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 21, \quad \ell = 2}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(21);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2^2q^{2} + -3q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(21);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1355*2^2 + O(2^20)q^{2} + 3q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(21);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(21);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 23, \quad \ell = 2}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(23);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + zq^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{0}&0\\0&u(z)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2q^{2} + -5q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(23);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + zq^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{0}&0\\0&u(z)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -91733*2 + O(2^20)q^{2} + -283929 + O(2^20)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(23);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + zq^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(23);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(23);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 25, \quad \ell = 2}$ \quad (3 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(25);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(25);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(25);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 27, \quad \ell = 2}$ \quad (3 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(27);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -9*2$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(27);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(27);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 29, \quad \ell = 2}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(29);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(112943*2^-2 + O(2^17))*a + 30177 + O(2^17)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 7625*2^6 + O(2^20)x + 5849*2^6 + O(2^20)$.
The slope of $f_0$ is $3$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(29);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(29);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(29);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 31, \quad \ell = 2}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(31);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + zq^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{0}&0\\0&u(z)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -210503*2 + O(2^20)q^{2} + 421003*2 + O(2^21)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(31);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + zq^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{0}&0\\0&u(z)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -58155*2 + O(2^20)q^{2} + 11495*2^2 + O(2^17)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(31);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + zq^{2} + \cdots \in S_{4}(\Gamma_0(31);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(31);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 33, \quad \ell = 2}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(33);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -33595*2^3 + O(2^20)q^{2} + -3q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $3$.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(33);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 10631*2^3 + O(2^20)q^{2} + 3q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $3$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(33);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(33);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(33);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(33);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 35, \quad \ell = 2}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(35);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -2^2*a + 17q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -2^3x + 7*2$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(35);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -(107937*2^2 + O(2^20))*a + 178075 + O(2^20)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -107937*2^2 + O(2^20)x + 89031*2 + O(2^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(35);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(35);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 37, \quad \ell = 2}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(37);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + (2503*2^3 + O(2^17))*a + 57997 + O(2^17)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -38327*2^2 + O(2^20)x + 1619*2 + O(2^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(37);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -(6451*2^2 + O(2^17))*a - 42319 + O(2^18)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -239867*2 + O(2^20)x + 82871*2 + O(2^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + zq^{3} + \cdots \in S_{4}(\Gamma_0(37);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(37);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + zq^{3} + \cdots \in S_{4}(\Gamma_0(37);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 39, \quad \ell = 2}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(39);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -3q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(39);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 5405*2^3 + O(2^20)q^{2} + 3q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $3$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(39);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + \cdots \in S_{4}(\Gamma_0(39);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 41, \quad \ell = 2}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(41);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (31327*2^-6 + O(2^14))*a^3 + (16127*2^-4 + O(2^14))*a^2 + (6699*2^-2 + O(2^14))*a + 23*2^5 + O(2^14)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{4} + 7961*2^4 + O(2^20)x^{3} + -5623*2^6 + O(2^20)x^{2} + 475*2^7 + O(2^20)x + 145*2^9 + O(2^20)$.
The slope of $f_0$ is $9/4$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(41);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(41);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(41);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 43, \quad \ell = 2}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(43);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -(2533*2 + O(2^17))*a + 17*2 + O(2^17)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 216203*2 + O(2^20)x + -87679*2 + O(2^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(43);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -(89971*2^-1 + O(2^17))*a^3 + (1121*2^2 + O(2^17))*a^2 - (5067*2^2 + O(2^17))*a + 10999*2^2 + O(2^17)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{4} + -29505*2^3 + O(2^20)x^{3} + -199705*2 + O(2^20)x^{2} + -52211*2^3 + O(2^20)x + 37431*2^2 + O(2^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + zq^{3} + \cdots \in S_{4}(\Gamma_0(43);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + z^2q^{3} + \cdots \in S_{4}(\Gamma_0(43);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 45, \quad \ell = 2}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(45);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2^2q^{2} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(45);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(45);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(45);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(45);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 47, \quad \ell = 2}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + a^2q^{3} + \cdots \in S_{4}(\Gamma_0(47);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^6q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^9)\omega^{0}&0\\0&u(z^6)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(342759*2^-1 + O(2^19))*a - 218379 + O(2^19)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -171377*2 + O(2^20)x + -109189*2^2 + O(2^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + a^2q^{3} + \cdots \in S_{4}(\Gamma_0(47);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{2} + x + =0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^6q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^9)\omega^{0}&0\\0&u(z^6)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (29517*2^-1 + O(2^16))*a + 21745 + O(2^16)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -46513*2 + O(2^20)x + 111751*2^2 + O(2^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(47);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^3q^{2} + z^5q^{3} + \cdots \in S_{4}(\Gamma_0(47);\overline{\mathbf{F}}_{2}).$
Also,~$z$ satisfies the equation $x^{4} + x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(47);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 49, \quad \ell = 2}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{4}(\Gamma_0(49);\overline{\mathbf{F}}_{2}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + O(2^20)*a + 19983*2^4 + O(2^20)q^{2} + -(1 + O(2^20))*a + 19983*2^4 + O(2^20)q^{3} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 12785*2^5 + O(2^20)x + 17615*2^3 + O(2^20)$.
The slope of $f_0$ is $4$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(49);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(49);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{3} + \cdots \in S_{4}(\Gamma_0(49);\overline{\mathbf{F}}_{2}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(49);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + \cdots \in S_{4}(\Gamma_0(49);\overline{\mathbf{F}}_{2}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\setcounter{section}{2}
\section{$\mathbf{2 \leq N \leq 70, \quad \ell = 3}$}

\subsection{$\mathbf{N = 2, \quad \ell = 3}$ \quad (no forms)}

There are no forms to analyze.

\subsection{$\mathbf{N = 4, \quad \ell = 3}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + \cdots \in S_{6}(\Gamma_0(4);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 5, \quad \ell = 3}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 2q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(5);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 7, \quad \ell = 3}$ \quad (2 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(7);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(7);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 8, \quad \ell = 3}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{3} + q^{5} + \cdots \in S_{6}(\Gamma_0(8);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 10, \quad \ell = 3}$ \quad (3 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(10);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + 2*3q^{3} + 16q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(10);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + 8*3q^{3} + 16q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(10);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 11, \quad \ell = 3}$ \quad (3 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(11);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -5*3q^{3} + -16q^{4} + -19q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(11);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(2)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 768257473 + O(3^20)q^{2} + 96924388*3 + O(3^19)q^{3} + 1033139585 + O(3^20)q^{4} + -1697335037 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + 2q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(11);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 13, \quad \ell = 3}$ \quad (4 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + zq^{2} + q^{3} + z^7q^{4} + z^2q^{5} + \cdots \in S_{6}(\Gamma_0(13);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^3q^{2} + q^{3} + z^5q^{4} + z^6q^{5} + \cdots \in S_{6}(\Gamma_0(13);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(13);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(13);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 14, \quad \ell = 3}$ \quad (2 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(14);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(14);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 16, \quad \ell = 3}$ \quad (2 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + \cdots \in S_{6}(\Gamma_0(16);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{3} + q^{5} + \cdots \in S_{6}(\Gamma_0(16);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 17, \quad \ell = 3}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(17);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + q^{2} + -2*3^2q^{3} + -31q^{4} + -16q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(17);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + (828968360*3^-1 + O(3^19))*a + 397823729*3^-1 + O(3^19)q^{3} + -(1686638999 + O(3^20))*a + 172039336*3 + O(3^20)q^{4} + (1630537157 + O(3^20))*a - 76122263*3^2 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 1686638999 + O(3^20)x + -516118040 + O(3^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(17);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^5q^{2} + 2q^{3} + z^7q^{4} + z^2q^{5} + \cdots \in S_{6}(\Gamma_0(17);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 19, \quad \ell = 3}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(19);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 3/22*a^3 - 5*3/22*a^2 - 35*3/11*a + 181*3/11q^{3} + a^2 - 32q^{4} + -23/22*a^3 + 3^3/22*a^2 + 893/11*a - 56*3^3/11q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{4} + -3^2x^{3} + -8*3^2x^{2} + 86*3^2x + -380*3$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + 2q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(19);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(19);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(19);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 20, \quad \ell = 3}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(20);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 22, \quad \ell = 3}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(22);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -7*3q^{3} + 16q^{4} + 3^4q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(22);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + -372933190*3 + O(3^20)q^{3} + 16q^{4} + -51095218*3^3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(22);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(22);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(22);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 23, \quad \ell = 3}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + zq^{2} + z^7q^{4} + z^5q^{5} + \cdots \in S_{6}(\Gamma_0(23);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^5q^{2} + z^6q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{0}&0\\0&u(z^6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(14041298*3 + O(3^20))*a + 301865704*3 + O(3^20)q^{3} + (168495562 + O(3^20))*a - 135604087 + O(3^20)q^{4} + (84247781 + O(3^20))*a + 558530045*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -168495562 + O(3^20)x + 135604055 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^3q^{2} + z^5q^{4} + z^7q^{5} + \cdots \in S_{6}(\Gamma_0(23);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^7q^{2} + z^2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^6)\omega^{0}&0\\0&u(z^2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(14041298*3 + O(3^20))*a + 301865704*3 + O(3^20)q^{3} + (168495562 + O(3^20))*a - 135604087 + O(3^20)q^{4} + (84247781 + O(3^20))*a + 558530045*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -168495562 + O(3^20)x + 135604055 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^5q^{2} + z^7q^{4} + zq^{5} + \cdots \in S_{6}(\Gamma_0(23);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^5q^{2} + z^6q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{1}&0\\0&u(z^6)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (47741734*3 + O(3^19))*a - 108721528*3 + O(3^19)q^{3} + -(1041843502 + O(3^20))*a + 534064556 + O(3^20)q^{4} + (835880963 + O(3^20))*a - 162616748*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 1041843502 + O(3^20)x + -534064588 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(23);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + 2q^{3} + z^3q^{5} + \cdots \in S_{6}(\Gamma_0(23);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^5q^{2} + z^6q^{3} + z^7q^{4} + zq^{5} + \cdots \in S_{6}(\Gamma_0(23);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 25, \quad \ell = 3}$ \quad (7 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(25);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^6q^{2} + \cdots \in S_{6}(\Gamma_0(25);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + z^2q^{2} + \cdots \in S_{6}(\Gamma_0(25);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(25);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(25);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(25);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(25);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 26, \quad \ell = 3}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(26);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + -4q^{2} + 16q^{4} + -14q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

\item  Consider
$f = q + q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(26);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + 4q^{2} + aq^{3} + 16q^{4} + -3*a + 50q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3^2x + -64*3$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(26);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(26);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 28, \quad \ell = 3}$ \quad (2 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + \cdots \in S_{6}(\Gamma_0(28);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + q^{5} + \cdots \in S_{6}(\Gamma_0(28);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 29, \quad \ell = 3}$ \quad (7 forms)}

\begin{enumerate}
\item  Consider
$f = q + c^5q^{2} + c^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(29);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + cq^{2} + c^5q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^3)\omega^{0}&0\\0&u(c^5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (485908895*3 + O(3^20))*a + 47013614*3^3 + O(3^20)q^{3} + -(343712983 + O(3^20))*a + 205336370 + O(3^20)q^{4} + -(177711077*3^2 + O(3^20))*a - 1158569672 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 343712983 + O(3^20)x + -205336402 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + c^7q^{2} + c^5q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(29);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + c^3q^{2} + c^7q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c)\omega^{0}&0\\0&u(c^7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (485908895*3 + O(3^20))*a + 47013614*3^3 + O(3^20)q^{3} + -(343712983 + O(3^20))*a + 205336370 + O(3^20)q^{4} + -(177711077*3^2 + O(3^20))*a - 1158569672 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 343712983 + O(3^20)x + -205336402 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + cq^{2} + c^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(29);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + cq^{2} + c^5q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^3)\omega^{1}&0\\0&u(c^5)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(35132759*3 + O(3^19))*a - 104509003*3 + O(3^19)q^{3} + -(568806134 + O(3^20))*a - 842362495 + O(3^20)q^{4} + (56814136*3^3 + O(3^20))*a - 636425824 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 568806134 + O(3^20)x + 842362463 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + cq^{2} + q^{3} + c^7q^{4} + c^7q^{5} + \cdots \in S_{6}(\Gamma_0(29);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + c^3q^{2} + q^{3} + c^5q^{4} + c^5q^{5} + \cdots \in S_{6}(\Gamma_0(29);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + a^7q^{2} + 2q^{3} + a^16q^{4} + a^22q^{5} + \cdots \in S_{6}(\Gamma_0(29);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + cq^{2} + c^5q^{3} + c^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(29);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 31, \quad \ell = 3}$ \quad (9 forms)}

\begin{enumerate}
\item  Consider
$f = q + zq^{2} + z^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + zq^{2} + zq^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^7)\omega^{1}&0\\0&u(z)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(59399854*3 + O(3^18))*a - 60392801*3 + O(3^18)q^{3} + -(26993603 + O(3^20))*a + 507311864 + O(3^20)q^{4} + (60958945*3 + O(3^19))*a - 437126999 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 26993603 + O(3^20)x + -507311896 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^5q^{2} + z^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + zq^{2} + zq^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^7)\omega^{0}&0\\0&u(z)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(61242059*3 + O(3^18))*a + 14785886*3^2 + O(3^18)q^{3} + -(839661361 + O(3^20))*a + 332501318 + O(3^20)q^{4} + -(150252932*3 + O(3^19))*a + 95195480 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 839661361 + O(3^20)x + -332501350 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^546q^{2} + 2q^{3} + z^455q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^28q^{2} + q^{3} + z^588q^{4} + z^532q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^84q^{2} + q^{3} + z^308q^{4} + z^140q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^182q^{2} + 2q^{3} + z^637q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^252q^{2} + q^{3} + z^196q^{4} + z^420q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + zq^{2} + zq^{3} + z^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(31);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 32, \quad \ell = 3}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{5} + \cdots \in S_{6}(\Gamma_0(32);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + -82q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

\item  Consider
$f = q + q^{5} + \cdots \in S_{6}(\Gamma_0(32);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{3} + 46q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -256*3$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(32);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(32);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 34, \quad \ell = 3}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(34);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + aq^{3} + 16q^{4} + 4*3^-1*a - 14q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 2*3x + -68*3^2$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(34);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + aq^{3} + 16q^{4} + -(1309201658*3^-1 + O(3^19))*a + 385290877 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -142520176*3 + O(3^20)x + 132689539*3^2 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(34);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(34);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(34);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(34);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 35, \quad \ell = 3}$ \quad (8 forms)}

\begin{enumerate}
\item  Consider
$f = q + zq^{2} + z^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^5q^{2} + z^7q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z)\omega^{0}&0\\0&u(z^7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -3*a + 3q^{3} + a - 16q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -1x + -16$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^3q^{2} + z^5q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^7q^{2} + z^5q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^3)\omega^{0}&0\\0&u(z^5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -3*a + 3q^{3} + a - 16q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -1x + -16$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^5q^{2} + z^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^5q^{2} + z^7q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z)\omega^{1}&0\\0&u(z^7)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(145944434*3 + O(3^19))*a + 10184609*3^3 + O(3^19)q^{3} + -(76109761 + O(3^20))*a - 1069901098 + O(3^20)q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 76109761 + O(3^20)x + 1069901066 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^6q^{2} + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^5q^{2} + z^7q^{3} + z^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(35);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 37, \quad \ell = 3}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(37);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + -261495886 + O(3^20)q^{2} + -24617477*3^2 + O(3^19)q^{3} + -1072438657 + O(3^20)q^{4} + -411596752 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + q^{2} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(37);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -(1699627921*3^-1 + O(3^19))*a + 1522894744*3^-1 + O(3^19)q^{3} + -(84265174 + O(3^20))*a + 257270885*3 + O(3^20)q^{4} + -(1551044939 + O(3^20))*a - 529209481*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 84265174 + O(3^20)x + -771812687 + O(3^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(37);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (5386375*3^2 + O(3^19))*a^3 + (162643129*3 + O(3^19))*a^2 - (15843080*3 + O(3^19))*a - 172566433*3 + O(3^19)q^{3} + a^2 - 32q^{4} + -(844412614 + O(3^20))*a^3 - (150025085*3^2 + O(3^20))*a^2 - (44174068 + O(3^20))*a + 14467880*3^5 + O(3^21)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{4} + 7178639*3^3 + O(3^20)x^{3} + -227042*3 + O(3^20)x^{2} + 171993760*3^2 + O(3^20)x + -9566960*3 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^13q^{2} + q^{3} + z^72q^{4} + z^54q^{5} + \cdots \in S_{6}(\Gamma_0(37);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{4} + 2x^{3} + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(37);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^14q^{2} + 2q^{3} + z^21q^{4} + z^8q^{5} + \cdots \in S_{6}(\Gamma_0(37);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 38, \quad \ell = 3}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(38);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -2*3q^{3} + 16q^{4} + 31q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(38);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(2)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + -9661139*3 + O(3^20)q^{3} + 16q^{4} + 1167061640 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(38);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(38);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(38);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(38);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 40, \quad \ell = 3}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{5} + \cdots \in S_{6}(\Gamma_0(40);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + -2*3^2q^{3} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + q^{5} + \cdots \in S_{6}(\Gamma_0(40);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{3} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 4*3x + -160*3$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(40);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{5} + \cdots \in S_{6}(\Gamma_0(40);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 41, \quad \ell = 3}$ \quad (10 forms)}

\begin{enumerate}
\item  Consider
$f = q + z^420q^{2} + z^504q^{4} + z^588q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^2q^{2} + z^14q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^12)\omega^{0}&0\\0&u(z^14)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (361506644*3 + O(3^20))*a^2 + (123336743*3 + O(3^20))*a - 142810786*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + -(4062758*3^3 + O(3^20))*a^2 + (1416069860 + O(3^20))*a + 171085594 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -1732513015 + O(3^20)x^{2} + -369378572 + O(3^20)x + -261992342 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^532q^{2} + z^56q^{4} + z^308q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^6q^{2} + z^16q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^10)\omega^{0}&0\\0&u(z^16)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (361506644*3 + O(3^20))*a^2 + (123336743*3 + O(3^20))*a - 142810786*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + -(4062758*3^3 + O(3^20))*a^2 + (1416069860 + O(3^20))*a + 171085594 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -1732513015 + O(3^20)x^{2} + -369378572 + O(3^20)x + -261992342 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^140q^{2} + z^168q^{4} + z^196q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^18q^{2} + z^22q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^4)\omega^{0}&0\\0&u(z^22)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (361506644*3 + O(3^20))*a^2 + (123336743*3 + O(3^20))*a - 142810786*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + -(4062758*3^3 + O(3^20))*a^2 + (1416069860 + O(3^20))*a + 171085594 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -1732513015 + O(3^20)x^{2} + -369378572 + O(3^20)x + -261992342 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^2q^{2} + z^18q^{4} + z^8q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^2q^{2} + z^14q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^12)\omega^{1}&0\\0&u(z^14)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(62479930*3^2 + O(3^19))*a^2 - (176107054*3 + O(3^19))*a + 131184580*3 + O(3^19)q^{3} + a^2 - 32q^{4} + -(144889612*3^2 + O(3^20))*a^2 - (775411618 + O(3^20))*a - 524825446 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 802283635 + O(3^20)x^{2} + -1403525993 + O(3^20)x + 963205139 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^546q^{2} + q^{3} + z^91q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^182q^{2} + q^{3} + z^273q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{6} + 2x^{4} + x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + z^14q^{3} + z^18q^{4} + z^8q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^8q^{2} + 2q^{3} + z^22q^{4} + z^5q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(41);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 43, \quad \ell = 3}$ \quad (10 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1151662513 + O(3^20)q^{2} + -66064810*3 + O(3^19)q^{3} + -556198801 + O(3^20)q^{4} + -1501437224 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1385818214 + O(3^20)q^{2} + -109835833*3 + O(3^19)q^{3} + -1686754798 + O(3^20)q^{4} + 147075986 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^2q^{2} + z^5q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^6q^{2} + z^2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^6)\omega^{0}&0\\0&u(z^2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (36548588*3 + O(3^18))*a - 7430932*3^2 + O(3^18)q^{3} + (21441881*3^3 + O(3^20))*a + 30681451*3 + O(3^20)q^{4} + -(41934607 + O(3^17))*a - 63117281 + O(3^17)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -21441881*3^3 + O(3^20)x + -92044385 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^2q^{2} + z^3q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^2q^{2} + z^6q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{1}&0\\0&u(z^6)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (1140656*3 + O(3^18))*a + 12331391*3^2 + O(3^18)q^{3} + (50344022*3^2 + O(3^20))*a - 451817726*3 + O(3^20)q^{4} + -(41413471 + O(3^17))*a + 18344459 + O(3^17)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -50344022*3^2 + O(3^20)x + 1355453146 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^5q^{2} + q^{3} + z^6q^{4} + z^7q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + z^6q^{3} + z^3q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^5q^{2} + 2q^{3} + z^7q^{4} + z^7q^{5} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(43);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 44, \quad \ell = 3}$ \quad (3 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(44);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(44);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + \cdots \in S_{6}(\Gamma_0(44);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 46, \quad \ell = 3}$ \quad (8 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + 294421781*3 + O(3^20)q^{3} + 16q^{4} + 19362385*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + -4q^{2} + aq^{3} + 16q^{4} + (49875845*3 + O(3^19))*a + 446738977 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 32713531*3^3 + O(3^20)x + -34110940*3 + O(3^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 4q^{2} + -11817551*3^4 + O(3^20)q^{3} + 16q^{4} + 393171758 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $4$.

\item  Consider
$f = q + q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + 127609667*3 + O(3^20)q^{3} + 16q^{4} + 142014293*3 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + zq^{5} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(46);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 47, \quad \ell = 3}$ \quad (10 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1653526378 + O(3^20)q^{2} + -130264028*3 + O(3^20)q^{3} + -1693489162 + O(3^20)q^{4} + 402185927*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + e^17q^{2} + e^15q^{4} + e^7q^{5} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + e^4q^{2} + e^25q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(e)\omega^{0}&0\\0&u(e^25)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (509900633*3 + O(3^20))*a^2 - (118738282*3^2 + O(3^20))*a + 236689189*3 + O(3^20)q^{3} + a^2 - 32q^{4} + -(164358022*3^2 + O(3^20))*a^2 - (879986902 + O(3^20))*a - 748983769 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 46652102 + O(3^20)x^{2} + -32551807*3 + O(3^20)x + 419601076 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(2)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 314431748 + O(3^20)q^{2} + 1138795*3 + O(3^15)q^{3} + -1231616200 + O(3^20)q^{4} + 1393093*3 + O(3^15)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + e^4q^{2} + e^15q^{4} + e^20q^{5} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + e^4q^{2} + e^25q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(e)\omega^{1}&0\\0&u(e^25)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(681929*3 + O(3^15))*a^2 - (1580135*3 + O(3^15))*a - 2299130*3 + O(3^15)q^{3} + a^2 - 32q^{4} + (248440*3^3 + O(3^15))*a^2 + (2601347 + O(3^15))*a - 3114497 + O(3^15)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -1494047123 + O(3^20)x^{2} + -462015928*3 + O(3^20)x + 1621688861 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + b^5q^{2} + 2q^{3} + b^7q^{4} + b^3q^{5} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + e^4q^{2} + e^25q^{3} + e^15q^{4} + e^20q^{5} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(47);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 49, \quad \ell = 3}$ \quad (12 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 11q^{2} + 89q^{4} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

\item  Consider
$f = q + q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + -2q^{2} + -1/2*a - 1q^{3} + -28q^{4} + -3/2*a - 3q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 4x + -2492$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + z^3q^{2} + 2q^{3} + z^5q^{4} + z^7q^{5} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^3q^{2} + q^{3} + z^5q^{4} + z^3q^{5} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + zq^{2} + 2q^{3} + z^7q^{4} + z^5q^{5} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + zq^{2} + q^{3} + z^7q^{4} + zq^{5} + \cdots \in S_{6}(\Gamma_0(49);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 50, \quad \ell = 3}$ \quad (7 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(50);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -2*3q^{3} + 16q^{4} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(50);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + -8*3q^{3} + 16q^{4} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(50);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(50);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(50);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(50);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(50);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 52, \quad \ell = 3}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{5} + \cdots \in S_{6}(\Gamma_0(52);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 26369093*3^2 + O(3^20)q^{3} + -454783301 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 2q^{5} + \cdots \in S_{6}(\Gamma_0(52);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{3} + -(47205994*3 + O(3^19))*a - 258276565 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 79107275*3 + O(3^20)x + -129148660*3 + O(3^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + \cdots \in S_{6}(\Gamma_0(52);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(52);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 53, \quad \ell = 3}$ \quad (9 forms)}

\begin{enumerate}
\item  Consider
$f = q + e^17q^{2} + e^15q^{4} + e^21q^{5} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + e^4q^{2} + e^9q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(e^17)\omega^{0}&0\\0&u(e^9)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (5331601*3 + O(3^17))*a^2 + (8579489*3 + O(3^17))*a - 5093699*3^2 + O(3^17)q^{3} + a^2 - 32q^{4} + -(22609735 + O(3^18))*a^2 - (1849567*3^4 + O(3^18))*a + 12349451*3 + O(3^18)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -1111115428 + O(3^20)x^{2} + -32142394*3^3 + O(3^20)x + 1484898871 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + (671653523*3^-3 + O(3^17))*a + 626262563*3^-3 + O(3^17)q^{3} + -(394255091 + O(3^20))*a + 446556856*3 + O(3^20)q^{4} + (1060874695*3^-2 + O(3^18))*a - 321686045*3^-2 + O(3^18)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 394255091 + O(3^20)x + -1339670600 + O(3^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 701788618 + O(3^20)q^{2} + 12813718*3^2 + O(3^18)q^{3} + -198717412 + O(3^20)q^{4} + -12944290*3^2 + O(3^18)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + e^4q^{2} + e^15q^{4} + e^8q^{5} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + e^4q^{2} + e^9q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(e^17)\omega^{1}&0\\0&u(e^9)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(18288626*3^2 + O(3^18))*a^2 - (56043461*3 + O(3^18))*a + 284704*3^5 + O(3^18)q^{3} + a^2 - 32q^{4} + (192596398 + O(3^18))*a^2 + (18249620*3^2 + O(3^18))*a + 30613450*3 + O(3^18)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -1067853935 + O(3^20)x^{2} + 50270663*3^2 + O(3^20)x + 433715999 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + eq^{2} + q^{3} + e^21q^{4} + e^6q^{5} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + e^8q^{2} + 2q^{3} + e^22q^{4} + e^7q^{5} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + e^4q^{2} + e^9q^{3} + e^15q^{4} + e^8q^{5} + \cdots \in S_{6}(\Gamma_0(53);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 55, \quad \ell = 3}$ \quad (12 forms)}

\begin{enumerate}
\item  Consider
$f = q + zq^{2} + z^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^5q^{2} + z^6q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{0}&0\\0&u(z^6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(149963012*3 + O(3^19))*a - 86865929*3 + O(3^19)q^{3} + (101575171 + O(3^20))*a + 1064982128 + O(3^20)q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -101575171 + O(3^20)x + -1064982160 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -389782084*3 + O(3^20)q^{2} + -98318002*3 + O(3^19)q^{3} + 1917043798 + O(3^21)q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -(913249939*3^-1 + O(3^19))*a + 678332182*3^-1 + O(3^19)q^{3} + (1067771090 + O(3^20))*a + 520777885*3 + O(3^20)q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -1067771090 + O(3^20)x + -1562333687 + O(3^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -172888007*3 + O(3^20)q^{2} + 423607919*3 + O(3^20)q^{3} + 981427216 + O(3^21)q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^5q^{2} + z^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + z^5q^{2} + z^6q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^2)\omega^{1}&0\\0&u(z^6)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (11697022*3 + O(3^19))*a - 178316809*3 + O(3^19)q^{3} + (263260781 + O(3^20))*a + 163222646 + O(3^20)q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -263260781 + O(3^20)x + -163222678 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + -1452883384 + O(3^20)q^{2} + 54271223*3^2 + O(3^19)q^{3} + 655483784 + O(3^20)q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^18q^{2} + q^{3} + z^6q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + q^{3} + z^18q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^6q^{2} + q^{3} + z^2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + zq^{2} + 2q^{3} + z^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^5q^{2} + z^6q^{3} + z^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(55);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 56, \quad \ell = 3}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{5} + \cdots \in S_{6}(\Gamma_0(56);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2*3q^{3} + 4q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{5} + \cdots \in S_{6}(\Gamma_0(56);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(2)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 10*3q^{3} + 32q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{5} + \cdots \in S_{6}(\Gamma_0(56);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{3} + 3*a + 50q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 2*3x + -112*3$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{5} + \cdots \in S_{6}(\Gamma_0(56);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 33868382*3^2 + O(3^20)q^{3} + -1524077204 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(56);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(56);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 58, \quad \ell = 3}$ \quad (11 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 4q^{2} + -53470127*3^3 + O(3^20)q^{3} + 16q^{4} + 163265525*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $3$.

\item  Consider
$f = q + 2q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -347715772*3 + O(3^20)q^{3} + 16q^{4} + -95531146*3 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -445443440*3 + O(3^20)q^{3} + 16q^{4} + 558702869 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + 325654298*3 + O(3^20)q^{3} + 16q^{4} + -49606112*3 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + 2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(2)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + 252497056*3 + O(3^20)q^{3} + 16q^{4} + 203645182 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + -4q^{2} + aq^{3} + 16q^{4} + (185620919*3 + O(3^19))*a + 155168791*3 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3085555*3^5 + O(3^20)x + 577826741*3 + O(3^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(58);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 59, \quad \ell = 3}$ \quad (13 forms)}

\begin{enumerate}
\item  Consider
$f = q + z^46848q^{2} + z^55876q^{4} + z^3904q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + c^71q^{2} + c^48q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^194)\omega^{0}&0\\0&u(c^48)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(514302740*3 + O(3^20))*a^4 - (47575549*3^3 + O(3^20))*a^3 + (43334773*3 + O(3^20))*a^2 + (576300484*3 + O(3^20))*a + 125419504*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + (168545018*3^2 + O(3^20))*a^4 + (75663416 + O(3^20))*a^3 + (47506472 + O(3^20))*a^2 - (344288051*3 + O(3^20))*a - 926330182 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -229419539*3 + O(3^20)x^{4} + 580411007*3 + O(3^20)x^{3} + 612744433 + O(3^20)x^{2} + -726172448 + O(3^20)x + -622416646 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^8296q^{2} + z^30500q^{4} + z^35136q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + c^155q^{2} + c^190q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^52)\omega^{0}&0\\0&u(c^190)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(514302740*3 + O(3^20))*a^4 - (47575549*3^3 + O(3^20))*a^3 + (43334773*3 + O(3^20))*a^2 + (576300484*3 + O(3^20))*a + 125419504*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + (168545018*3^2 + O(3^20))*a^4 + (75663416 + O(3^20))*a^3 + (47506472 + O(3^20))*a^2 - (344288051*3 + O(3^20))*a - 926330182 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -229419539*3 + O(3^20)x^{4} + 580411007*3 + O(3^20)x^{3} + 612744433 + O(3^20)x^{2} + -726172448 + O(3^20)x + -622416646 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^15616q^{2} + z^38308q^{4} + z^20984q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + c^185q^{2} + c^16q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^226)\omega^{0}&0\\0&u(c^16)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(514302740*3 + O(3^20))*a^4 - (47575549*3^3 + O(3^20))*a^3 + (43334773*3 + O(3^20))*a^2 + (576300484*3 + O(3^20))*a + 125419504*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + (168545018*3^2 + O(3^20))*a^4 + (75663416 + O(3^20))*a^3 + (47506472 + O(3^20))*a^2 - (344288051*3 + O(3^20))*a - 926330182 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -229419539*3 + O(3^20)x^{4} + 580411007*3 + O(3^20)x^{3} + 612744433 + O(3^20)x^{2} + -726172448 + O(3^20)x + -622416646 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^22448q^{2} + z^49532q^{4} + z^11712q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + c^213q^{2} + c^144q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^98)\omega^{0}&0\\0&u(c^144)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(514302740*3 + O(3^20))*a^4 - (47575549*3^3 + O(3^20))*a^3 + (43334773*3 + O(3^20))*a^2 + (576300484*3 + O(3^20))*a + 125419504*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + (168545018*3^2 + O(3^20))*a^4 + (75663416 + O(3^20))*a^3 + (47506472 + O(3^20))*a^2 - (344288051*3 + O(3^20))*a - 926330182 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -229419539*3 + O(3^20)x^{4} + 580411007*3 + O(3^20)x^{3} + 612744433 + O(3^20)x^{2} + -726172448 + O(3^20)x + -622416646 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^24888q^{2} + z^32452q^{4} + z^46360q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + c^223q^{2} + c^86q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^156)\omega^{0}&0\\0&u(c^86)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(514302740*3 + O(3^20))*a^4 - (47575549*3^3 + O(3^20))*a^3 + (43334773*3 + O(3^20))*a^2 + (576300484*3 + O(3^20))*a + 125419504*3^2 + O(3^20)q^{3} + a^2 - 32q^{4} + (168545018*3^2 + O(3^20))*a^4 + (75663416 + O(3^20))*a^3 + (47506472 + O(3^20))*a^2 - (344288051*3 + O(3^20))*a - 926330182 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -229419539*3 + O(3^20)x^{4} + 580411007*3 + O(3^20)x^{3} + 612744433 + O(3^20)x^{2} + -726172448 + O(3^20)x + -622416646 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + c^71q^{2} + c^229q^{4} + c^137q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{5} + 2x + =0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + c^71q^{2} + c^48q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(c^194)\omega^{1}&0\\0&u(c^48)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (166918577*3 + O(3^19))*a^4 - (107477639*3 + O(3^19))*a^3 - (27977363*3^2 + O(3^19))*a^2 + (111122560*3 + O(3^19))*a + 175854971*3 + O(3^19)q^{3} + a^2 - 32q^{4} + -(13334003*3^2 + O(3^20))*a^4 + (506287667 + O(3^20))*a^3 + (226975510 + O(3^20))*a^2 - (237537488*3 + O(3^20))*a - 789117095 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + 324487241*3 + O(3^20)x^{4} + 116829959*3 + O(3^20)x^{3} + 363803114 + O(3^20)x^{2} + -683914757 + O(3^20)x + 744462514 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^36905q^{2} + 2q^{3} + z^51667q^{4} + z^44286q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^51667q^{2} + 2q^{3} + z^36905q^{4} + z^14762q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^7381q^{2} + q^{3} + z^51667q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^22143q^{2} + q^{3} + z^36905q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + a^2q^{2} + q^{3} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + c^71q^{2} + c^48q^{3} + c^229q^{4} + c^137q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{5} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + d^14q^{2} + 2q^{3} + d^21q^{4} + d^19q^{5} + \cdots \in S_{6}(\Gamma_0(59);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 61, \quad \ell = 3}$ \quad (11 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1599162835 + O(3^20)q^{2} + 90647756*3 + O(3^19)q^{3} + -1407332329 + O(3^20)q^{4} + -41465042*3^3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -808135042 + O(3^20)q^{2} + 163128800*3 + O(3^19)q^{3} + -166480027 + O(3^20)q^{4} + -37734250*3^2 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + d^17q^{2} + d^15q^{4} + d^10q^{5} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + d^17q^{2} + d^21q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(d^5)\omega^{1}&0\\0&u(d^21)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(2194184*3^4 + O(3^18))*a^2 + (35077376*3 + O(3^18))*a - 28512515*3 + O(3^18)q^{3} + a^2 - 32q^{4} + -(72789638 + O(3^18))*a^2 - (110602664 + O(3^18))*a + 151534504 + O(3^18)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -779482108 + O(3^20)x^{2} + 12823322*3^4 + O(3^20)x + 1719292888 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + d^4q^{2} + d^15q^{4} + d^23q^{5} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + d^17q^{2} + d^21q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(d^5)\omega^{0}&0\\0&u(d^21)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(50470253*3 + O(3^18))*a^2 + (7236025*3^2 + O(3^18))*a + 8308406*3^2 + O(3^18)q^{3} + a^2 - 32q^{4} + -(114074392 + O(3^18))*a^2 - (124065863 + O(3^18))*a - 129268900 + O(3^18)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -1183229405 + O(3^20)x^{2} + -65561014*3^2 + O(3^20)x + -1053207253 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + cq^{2} + 2q^{3} + c^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + c^2q^{2} + q^{3} + c^7q^{5} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + d^5q^{2} + q^{3} + d^6q^{4} + d^16q^{5} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + d^17q^{2} + 2q^{3} + d^15q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + d^17q^{2} + d^21q^{3} + d^15q^{4} + d^10q^{5} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(61);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 62, \quad \ell = 3}$ \quad (9 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + -4q^{2} + -78220336*3^2 + O(3^20)q^{3} + 16q^{4} + 1407966041 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + 325496924*3 + O(3^20)q^{3} + 16q^{4} + 511267598*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + q^{4} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -510084745*3 + O(3^20)q^{3} + 16q^{4} + -152279534*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + 4q^{2} + aq^{3} + 16q^{4} + (157788533*3 + O(3^19))*a - 417681821 + O(3^19)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -207667840*3 + O(3^20)x + 502161137*3 + O(3^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + z^6q^{5} + \cdots \in S_{6}(\Gamma_0(62);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 64, \quad \ell = 3}$ \quad (8 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{5} + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 82q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

\item  Consider
$f = q + 2q^{5} + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{3} + -46q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -256*3$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{3} + q^{5} + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{5} + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(64);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 65, \quad \ell = 3}$ \quad (12 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 5q^{2} + 2*3q^{3} + -7q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 13758946*3^2 + O(3^20)q^{2} + -144843037*3 + O(3^20)q^{3} + -13388777987 + O(3^22)q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + a^5q^{2} + a^7q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + aq^{2} + a^2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(a^6)\omega^{0}&0\\0&u(a^2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (166345072*3 + O(3^20))*a + 375315836*3 + O(3^20)q^{3} + (238194770 + O(3^20))*a - 738647491 + O(3^20)q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -238194770 + O(3^20)x + 738647459 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 58971128*3^3 + O(3^20)q^{2} + 510193673*3 + O(3^21)q^{3} + 39783429013 + O(3^23)q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 177024910 + O(3^20)q^{2} + -34429429*3 + O(3^19)q^{3} + -1235135131 + O(3^20)q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + aq^{2} + a^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + aq^{2} + a^2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(a^6)\omega^{1}&0\\0&u(a^2)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (180951337*3 + O(3^19))*a - 23569679*3 + O(3^19)q^{3} + (347870677 + O(3^20))*a + 32218088 + O(3^20)q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -347870677 + O(3^20)x + -32218120 + O(3^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + cq^{2} + 2q^{3} + c^21q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + aq^{2} + a^2q^{3} + a^7q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(65);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 67, \quad \ell = 3}$ \quad (12 forms)}

\begin{enumerate}
\item  Consider
$f = q + b^2q^{2} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + b^6q^{2} + b^2q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(b^6)\omega^{0}&0\\0&u(b^2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (90220453*3 + O(3^19))*a^3 - (25678615*3^2 + O(3^19))*a^2 - (21488869*3^2 + O(3^19))*a + 6176140*3^3 + O(3^19)q^{3} + a^2 - 32q^{4} + -(801566359 + O(3^20))*a^3 - (96457579*3^2 + O(3^20))*a^2 - (58037176 + O(3^20))*a + 20846537*3^2 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{4} + 5459413*3^2 + O(3^20)x^{3} + -1355760613 + O(3^20)x^{2} + 6379591*3^2 + O(3^20)x + -837687920 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + bq^{2} + b^7q^{4} + b^7q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + b^5q^{2} + b^3q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(b^5)\omega^{0}&0\\0&u(b^3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(175848968*3 + O(3^19))*a + 167631758*3 + O(3^19)q^{3} + (84575302 + O(3^20))*a + 1493788952 + O(3^20)q^{4} + (1068538375 + O(3^20))*a - 8516467 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -84575302 + O(3^20)x + -1493788984 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + b^5q^{2} + b^7q^{4} + b^3q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + b^5q^{2} + b^3q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(b^5)\omega^{1}&0\\0&u(b^3)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(122064892*3 + O(3^19))*a + 62471720*3^2 + O(3^19)q^{3} + -(261898081 + O(3^20))*a - 412676854 + O(3^20)q^{4} + (1392873787 + O(3^20))*a - 278585045 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 261898081 + O(3^20)x + 412676822 + O(3^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -191169581 + O(3^20)q^{2} + 6715778*3 + O(3^19)q^{3} + -1488284107 + O(3^20)q^{4} + -1075677026 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 2q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1337552924 + O(3^20)q^{2} + -119441722*3 + O(3^19)q^{3} + 298773839 + O(3^20)q^{4} + -50094526 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + b^2q^{2} + b^6q^{3} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + c^2q^{2} + q^{3} + c^189q^{4} + c^223q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{5} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + 2q^{4} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + b^5q^{2} + b^3q^{3} + b^7q^{4} + b^3q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{2} + 2x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + c^14q^{2} + 2q^{3} + c^21q^{4} + c^23q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + =0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{4} + \cdots \in S_{6}(\Gamma_0(67);\overline{\mathbf{F}}_{3}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 68, \quad \ell = 3}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + \cdots \in S_{6}(\Gamma_0(68);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 114227840*3 + O(3^20)q^{3} + -228455686*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + \cdots \in S_{6}(\Gamma_0(68);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 90734957*3 + O(3^20)q^{3} + -35592890*3 + O(3^20)q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + q^{5} + \cdots \in S_{6}(\Gamma_0(68);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(68);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + \cdots \in S_{6}(\Gamma_0(68);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 70, \quad \ell = 3}$ \quad (9 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4q^{2} + -3q^{3} + 16q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + -4q^{2} + -3^2q^{3} + 16q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + q^{2} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + q^{2} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + 4q^{2} + aq^{3} + 16q^{4} + -25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3x + -280*3$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{2} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{2} + q^{3} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4q^{2} + 22429019*3 + O(3^20)q^{3} + 16q^{4} + 25q^{5} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + 2q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + q^{5} + \cdots \in S_{6}(\Gamma_0(70);\overline{\mathbf{F}}_{3}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\setcounter{section}{4}
\section{$\mathbf{2 \leq N \leq 22, \quad \ell = 5}$}

\subsection{$\mathbf{N = 2, \quad \ell = 5}$ \quad (2 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 2q^{3} + 4q^{4} + 4q^{6} + q^{7} + 3q^{8} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(2);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 4q^{3} + q^{4} + 4q^{6} + 3q^{7} + q^{8} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(2);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 3, \quad \ell = 5}$ \quad (3 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + 3q^{3} + 3q^{4} + 3q^{6} + q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(3);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 3q^{2} + q^{3} + 2q^{4} + 3q^{6} + 3q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(3);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 4q^{2} + 4q^{3} + 4q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(3);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 4, \quad \ell = 5}$ \quad (1 form)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{3} + 4q^{5} + 2q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(4);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 6, \quad \ell = 5}$ \quad (2 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{2} + 2q^{3} + 4q^{4} + q^{5} + q^{6} + 4q^{7} + 2q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(6);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + q^{3} + q^{4} + 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(6);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 7, \quad \ell = 5}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 2q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(7);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{2} + 3q^{3} + 3q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -2*a + 44q^{3} + -3*a + 86q^{4} + -2*5*a + 6*5^2q^{5} + 2*5^2*a - 428q^{6} + -343q^{7} + -33*a - 642q^{8} + -188*a + 121*5q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 3x + -214$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 2q^{4} + 2q^{6} + q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(7);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{2} + 3q^{3} + 3q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (38028478599507 + O(5^20))*a - 41676169300512 + O(5^20)q^{3} + (19902944725341 + O(5^20))*a + 5630890182319 + O(5^20)q^{4} + (3506612718082*5 + O(5^20))*a + 6082877787528*5 + O(5^20)q^{5} + (299488066526*5^3 + O(5^20))*a + 36045279120567 + O(5^20)q^{6} + 2401q^{7} + (36492112669963 + O(5^20))*a + 22881262179746 + O(5^20)q^{8} + (23216721506416 + O(5^20))*a - 1197718328164*5 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -19902944725341 + O(5^20)x + -5630890182831 + O(5^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 4q^{2} + 3q^{3} + 3q^{4} + q^{5} + 2q^{6} + 3q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(7);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^16q^{2} + z^8q^{3} + z^21q^{4} + q^{5} + q^{6} + 4q^{7} + z^20q^{8} + z^19q^{9} + \cdots \in S_{10}(\Gamma_0(7);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^8q^{2} + z^16q^{3} + z^9q^{4} + q^{5} + q^{6} + 4q^{7} + z^4q^{8} + z^23q^{9} + \cdots \in S_{10}(\Gamma_0(7);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 3q^{4} + 4q^{5} + q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(7);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 8, \quad \ell = 5}$ \quad (4 forms)}

\begin{enumerate}
\item  Consider
$f = q + 4q^{3} + q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(8);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{3} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 44q^{3} + 86*5q^{5} + -1224q^{7} + -251q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{3} + 3q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(8);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{3} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 68q^{3} + 302*5q^{5} + 10248q^{7} + -15059q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + 3q^{5} + 4q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(8);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{5} + q^{7} + 2q^{9} + \cdots \in S_{10}(\Gamma_0(8);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 9, \quad \ell = 5}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{4} + \cdots \in S_{8}(\Gamma_0(9);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{4} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{15}$. The form
$f_0 = q + aq^{2} + 232q^{4} + -16*aq^{5} + 52*5q^{7} + 104*aq^{8} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -72*5$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 3q^{4} + \cdots \in S_{10}(\Gamma_0(9);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{4} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{21}$. The form
$f_0 = q + -512q^{4} + -2516*5q^{7} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 4q^{2} + 3q^{4} + q^{7} + \cdots \in S_{8}(\Gamma_0(9);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + 3q^{7} + \cdots \in S_{10}(\Gamma_0(9);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 4q^{4} + 4q^{5} + 2q^{8} + \cdots \in S_{10}(\Gamma_0(9);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 11, \quad \ell = 5}$ \quad (11 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{2}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -6*a - 27q^{3} + -8*a - 84q^{4} + 4*5*a - 31*5q^{5} + 21*a - 264q^{6} + 82*a - 286q^{7} + -148*a - 352q^{8} + 36*a + 126q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 8x + -44$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^19q^{2} + z^11q^{3} + z^10q^{4} + 2q^{6} + z^9q^{7} + q^{8} + z^17q^{9} + \cdots \in S_{8}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + z^23q^{3} + z^10q^{4} + z^2q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^22)\omega^{1}&0\\0&u(z^2)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(14723740761909 + O(5^20))*a - 18457990057768 + O(5^20)q^{3} + -(3345838299947 + O(5^20))*a - 9712541111131 + O(5^20)q^{4} + (8567596081731*5 + O(5^20))*a + 772294121228*5 + O(5^20)q^{5} + -(28780860585289*5 + O(5^21))*a - 55798651659023 + O(5^21)q^{6} + (21074961207461 + O(5^20))*a - 46141526117749 + O(5^20)q^{7} + (1381066229287*5^2 + O(5^20))*a - 9284799530034 + O(5^20)q^{8} + -(41877807585508 + O(5^20))*a - 12433774789831 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 3345838299947 + O(5^20)x + 9712541111003 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^23q^{2} + z^7q^{3} + z^2q^{4} + 2q^{6} + z^21q^{7} + q^{8} + z^13q^{9} + \cdots \in S_{8}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^11q^{2} + z^19q^{3} + z^2q^{4} + z^10q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^14)\omega^{1}&0\\0&u(z^10)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(14723740761909 + O(5^20))*a - 18457990057768 + O(5^20)q^{3} + -(3345838299947 + O(5^20))*a - 9712541111131 + O(5^20)q^{4} + (8567596081731*5 + O(5^20))*a + 772294121228*5 + O(5^20)q^{5} + -(28780860585289*5 + O(5^21))*a - 55798651659023 + O(5^21)q^{6} + (21074961207461 + O(5^20))*a - 46141526117749 + O(5^20)q^{7} + (1381066229287*5^2 + O(5^20))*a - 9284799530034 + O(5^20)q^{8} + -(41877807585508 + O(5^20))*a - 12433774789831 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 3345838299947 + O(5^20)x + 9712541111003 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^13q^{2} + z^17q^{3} + z^22q^{4} + 2q^{6} + z^3q^{7} + 2q^{8} + z^5q^{9} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + z^23q^{3} + z^10q^{4} + z^2q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^22)\omega^{0}&0\\0&u(z^2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (30954438160334 + O(5^20))*a + 29996059870094 + O(5^20)q^{3} + (38466789639159 + O(5^20))*a + 46133815960931 + O(5^20)q^{4} + -(500823431929*5 + O(5^20))*a - 1075850406153*5 + O(5^20)q^{5} + (1074049443443*5^2 + O(5^20))*a - 15317408685538 + O(5^20)q^{6} + -(13425618042939 + O(5^20))*a + 7658704338523 + O(5^20)q^{7} + 8*5^2*a - 6688q^{8} + -(213904856443304 + O(5^21))*a + 87984304968211 + O(5^21)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -38466789639159 + O(5^20)x + -46133815961443 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^17q^{2} + z^13q^{3} + z^14q^{4} + 2q^{6} + z^15q^{7} + 2q^{8} + zq^{9} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^11q^{2} + z^19q^{3} + z^2q^{4} + z^10q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^14)\omega^{0}&0\\0&u(z^10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (30954438160334 + O(5^20))*a + 29996059870094 + O(5^20)q^{3} + (38466789639159 + O(5^20))*a + 46133815960931 + O(5^20)q^{4} + -(500823431929*5 + O(5^20))*a - 1075850406153*5 + O(5^20)q^{5} + (1074049443443*5^2 + O(5^20))*a - 15317408685538 + O(5^20)q^{6} + -(13425618042939 + O(5^20))*a + 7658704338523 + O(5^20)q^{7} + 8*5^2*a - 6688q^{8} + -(213904856443304 + O(5^21))*a + 87984304968211 + O(5^21)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -38466789639159 + O(5^20)x + -46133815961443 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (23313874539791 + O(5^20))*a - 25574079328894 + O(5^20)q^{3} + -(31889281285684 + O(5^20))*a + 41588807916519 + O(5^20)q^{4} + -(323394749909*5 + O(5^19))*a + 1895456261351*5 + O(5^19)q^{5} + -(38488229496563 + O(5^20))*a + 29471527877396 + O(5^20)q^{6} + (32286121866192 + O(5^20))*a + 578218072002 + O(5^20)q^{7} + (44537475795113 + O(5^20))*a - 45654127349829 + O(5^20)q^{8} + (42143088724888 + O(5^20))*a + 15292834551539 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 31889281285684 + O(5^20)x + -41588807917031 + O(5^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^7q^{2} + z^23q^{3} + z^10q^{4} + z^2q^{5} + 2q^{6} + z^21q^{7} + 4q^{8} + z^17q^{9} + \cdots \in S_{8}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^11q^{2} + z^19q^{3} + z^2q^{4} + z^10q^{5} + 2q^{6} + z^9q^{7} + 4q^{8} + z^13q^{9} + \cdots \in S_{8}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 4q^{4} + q^{5} + 2q^{8} + 2q^{9} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + 2q^{6} + 3q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + z^9q^{3} + z^7q^{4} + 4q^{5} + z^11q^{6} + 3q^{7} + zq^{8} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 12, \quad \ell = 5}$ \quad (3 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{3} + 2q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(12);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 2q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(2)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 27q^{3} + 54*5q^{5} + 1112q^{7} + 729q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{3} + q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(12);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 2q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -81q^{3} + 198*5q^{5} + 8576q^{7} + 6561q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{3} + 2q^{5} + 3q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(12);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 13, \quad \ell = 5}$ \quad (16 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{3} + 2q^{4} + 3q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 2q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2*5q^{2} + -73q^{3} + -28q^{4} + -59*5q^{5} + -146*5q^{6} + 1373q^{7} + -312*5q^{8} + 3142q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^16q^{2} + z^15q^{3} + z^23q^{4} + z^7q^{6} + z^16q^{7} + z^23q^{8} + \cdots \in S_{8}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^4q^{2} + z^3q^{3} + z^23q^{4} + z^7q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^17)\omega^{1}&0\\0&u(z^7)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (11502674914997 + O(5^20))*a + 25254120162651 + O(5^20)q^{3} + (27006403655779 + O(5^20))*a - 39794541661534 + O(5^20)q^{4} + -(5094180181544*5 + O(5^20))*a + 1166223899184*5 + O(5^20)q^{5} + (27281430480314 + O(5^20))*a + 16834204650468 + O(5^20)q^{6} + (21849841762476 + O(5^20))*a + 8841132770709*5 + O(5^20)q^{7} + -(31662478986696 + O(5^20))*a + 18367144334726 + O(5^20)q^{8} + (3403574756096*5 + O(5^20))*a + 4409996916592*5 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -27006403655779 + O(5^20)x + 39794541661406 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^8q^{2} + z^3q^{3} + z^19q^{4} + z^11q^{6} + z^8q^{7} + z^19q^{8} + \cdots \in S_{8}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^20q^{2} + z^15q^{3} + z^19q^{4} + z^11q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^13)\omega^{1}&0\\0&u(z^11)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (11502674914997 + O(5^20))*a + 25254120162651 + O(5^20)q^{3} + (27006403655779 + O(5^20))*a - 39794541661534 + O(5^20)q^{4} + -(5094180181544*5 + O(5^20))*a + 1166223899184*5 + O(5^20)q^{5} + (27281430480314 + O(5^20))*a + 16834204650468 + O(5^20)q^{6} + (21849841762476 + O(5^20))*a + 8841132770709*5 + O(5^20)q^{7} + -(31662478986696 + O(5^20))*a + 18367144334726 + O(5^20)q^{8} + (3403574756096*5 + O(5^20))*a + 4409996916592*5 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -27006403655779 + O(5^20)x + 39794541661406 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{3} + 3q^{4} + 4q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 2q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 48473920096*5^3 + O(5^20)q^{2} + 33839733665394 + O(5^20)q^{3} + -3822264514203637 + O(5^23)q^{4} + 9313460303297*5 + O(5^20)q^{5} + -981957704676*5^3 + O(5^21)q^{6} + -28930472149396 + O(5^20)q^{7} + 173951696696*5^3 + O(5^20)q^{8} + -201941289422572 + O(5^21)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^2q^{2} + z^9q^{3} + z^7q^{4} + z^11q^{6} + z^2q^{7} + zq^{8} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^20q^{2} + z^15q^{3} + z^19q^{4} + z^11q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^13)\omega^{0}&0\\0&u(z^11)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (14051061280893 + O(5^20))*a + 18335295278812 + O(5^20)q^{3} + (28458143436482 + O(5^20))*a + 2522525476144 + O(5^20)q^{4} + (163535058257*5 + O(5^20))*a - 1708064033718*5 + O(5^20)q^{5} + -(34842188435887 + O(5^20))*a + 42125701036933 + O(5^20)q^{6} + (4510139771486 + O(5^20))*a + 1182832232168*5^2 + O(5^20)q^{7} + -(4779729034919 + O(5^20))*a + 22621052342317 + O(5^20)q^{8} + -(1120777451344*5^2 + O(5^20))*a - 4113534249059*5 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -28458143436482 + O(5^20)x + -2522525476656 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^10q^{2} + z^21q^{3} + z^11q^{4} + z^7q^{6} + z^10q^{7} + z^5q^{8} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^4q^{2} + z^3q^{3} + z^23q^{4} + z^7q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^17)\omega^{0}&0\\0&u(z^7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (14051061280893 + O(5^20))*a + 18335295278812 + O(5^20)q^{3} + (28458143436482 + O(5^20))*a + 2522525476144 + O(5^20)q^{4} + (163535058257*5 + O(5^20))*a - 1708064033718*5 + O(5^20)q^{5} + -(34842188435887 + O(5^20))*a + 42125701036933 + O(5^20)q^{6} + (4510139771486 + O(5^20))*a + 1182832232168*5^2 + O(5^20)q^{7} + -(4779729034919 + O(5^20))*a + 22621052342317 + O(5^20)q^{8} + -(1120777451344*5^2 + O(5^20))*a - 4113534249059*5 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -28458143436482 + O(5^20)x + -2522525476656 + O(5^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^20q^{2} + z^15q^{3} + z^19q^{4} + z^11q^{5} + z^11q^{6} + z^20q^{7} + z^7q^{8} + \cdots \in S_{8}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^4q^{2} + z^3q^{3} + z^23q^{4} + z^7q^{5} + z^7q^{6} + z^4q^{7} + z^11q^{8} + \cdots \in S_{8}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{3} + 2q^{4} + 3q^{5} + 2q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 4q^{2} + 3q^{3} + 4q^{4} + 4q^{5} + 2q^{6} + q^{7} + 3q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^14q^{2} + z^3q^{3} + z^7q^{4} + q^{5} + z^17q^{6} + z^8q^{7} + z^13q^{8} + 4q^{9} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^22q^{2} + z^15q^{3} + z^11q^{4} + q^{5} + z^13q^{6} + z^16q^{7} + z^17q^{8} + 4q^{9} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + z^21q^{2} + z^11q^{3} + q^{4} + 4q^{5} + z^8q^{6} + 2q^{7} + z^9q^{8} + z^2q^{9} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^9q^{2} + z^7q^{3} + q^{4} + 4q^{5} + z^16q^{6} + 2q^{7} + z^21q^{8} + z^10q^{9} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 14, \quad \ell = 5}$ \quad (7 forms)}

\begin{enumerate}
\item  Consider
$f = q + 3q^{2} + 4q^{3} + 4q^{4} + 2q^{6} + 2q^{7} + 2q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(14);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 4q^{2} + 3q^{3} + q^{4} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{7}$. The form
$f_0 = q + 8q^{2} + -66q^{3} + 64q^{4} + -16*5^2q^{5} + -528q^{6} + -343q^{7} + 512q^{8} + 2169q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 3q^{2} + 2q^{3} + 4q^{4} + q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(14);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 3q^{3} + 4q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 8q^{2} + 44140511564472 + O(5^20)q^{3} + 64q^{4} + -3158975503474*5 + O(5^20)q^{5} + -28345634046724 + O(5^20)q^{6} + 343q^{7} + 512q^{8} + 228812860292847 + O(5^21)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 4q^{3} + q^{4} + q^{6} + 4q^{7} + 4q^{8} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(14);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 3q^{3} + 4q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -16q^{2} + -6q^{3} + 256q^{4} + 112*5q^{5} + 96q^{6} + -2401q^{7} + -4096q^{8} + -19647q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 3q^{3} + q^{4} + 2q^{6} + q^{7} + 4q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(14);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 4q^{2} + 3q^{3} + q^{4} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + -16q^{2} + aq^{3} + 256q^{4} + 21*5^-1*a - 6678*5^-1q^{5} + -16*aq^{6} + 2401q^{7} + -4096q^{8} + -14*a + 37893q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 14x + -57576$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 3q^{3} + 4q^{4} + 3q^{5} + q^{6} + 2q^{7} + 3q^{8} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(14);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 3q^{3} + 4q^{4} + q^{5} + 4q^{6} + 3q^{7} + 2q^{8} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(14);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{4} + 4q^{5} + 4q^{7} + q^{8} + 2q^{9} + \cdots \in S_{10}(\Gamma_0(14);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 16, \quad \ell = 5}$ \quad (7 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + 4q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(16);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -44q^{3} + 86*5q^{5} + 1224q^{7} + -251q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{3} + 2q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -68q^{3} + 302*5q^{5} + -10248q^{7} + -15059q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{3} + 4q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(16);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 4q^{3} + 3q^{5} + q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(16);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{5} + 4q^{7} + 2q^{9} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 2q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{3} + 4q^{5} + 3q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 17, \quad \ell = 5}$ \quad (20 forms)}

\begin{enumerate}
\item  Consider
$f = q + 3q^{2} + 3q^{3} + q^{4} + 4q^{6} + 3q^{7} + 4q^{8} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 2q^{3} + q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2q^{2} + 18q^{3} + -124q^{4} + -2*5q^{5} + -36q^{6} + -902q^{7} + 504q^{8} + -1863q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^69q^{2} + z^58q^{3} + z^63q^{4} + z^3q^{6} + z^121q^{7} + z^2q^{8} + z^104q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + z^120q^{3} + z^63q^{4} + z^95q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{1}&0\\0&u(z^95)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (36918740115879 + O(5^20))*a^2 + (26995823805512 + O(5^20))*a + 11766912184049 + O(5^20)q^{3} + a^2 - 128q^{4} + -(284390270399*5^2 + O(5^20))*a^2 - (5207406561004*5 + O(5^20))*a + 6598790002149*5 + O(5^20)q^{5} + -(45914080300817 + O(5^20))*a^2 + (602744931637*5^2 + O(5^20))*a + 25133829749243 + O(5^20)q^{6} + (717142301933*5^2 + O(5^20))*a^2 - (43015855277311 + O(5^20))*a - 10618306124786 + O(5^20)q^{7} + -(18371543705426 + O(5^20))*a^2 + (12405291764338 + O(5^20))*a + 9307391057342 + O(5^20)q^{8} + -(588118752198*5 + O(5^20))*a^2 - (42988300097032 + O(5^20))*a - 47451623568776 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 18371543705426 + O(5^20)x^{2} + -12405291764594 + O(5^20)x + -9307391057342 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + q^{4} + 3q^{7} + 4q^{8} + 3q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 4q^{2} + 4q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{2}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -26036239370422 + O(5^20)q^{2} + -697784769483*5^2 + O(5^20)q^{3} + -19607012948294 + O(5^20)q^{4} + -8579596927316*5 + O(5^20)q^{5} + 1643849681826*5^2 + O(5^20)q^{6} + -21107172468197 + O(5^20)q^{7} + -43956978422791 + O(5^20)q^{8} + 27216089709688 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^97q^{2} + z^42q^{3} + z^67q^{4} + z^15q^{6} + z^109q^{7} + z^10q^{8} + z^24q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^35q^{2} + z^104q^{3} + z^67q^{4} + z^103q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^21)\omega^{1}&0\\0&u(z^103)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (36918740115879 + O(5^20))*a^2 + (26995823805512 + O(5^20))*a + 11766912184049 + O(5^20)q^{3} + a^2 - 128q^{4} + -(284390270399*5^2 + O(5^20))*a^2 - (5207406561004*5 + O(5^20))*a + 6598790002149*5 + O(5^20)q^{5} + -(45914080300817 + O(5^20))*a^2 + (602744931637*5^2 + O(5^20))*a + 25133829749243 + O(5^20)q^{6} + (717142301933*5^2 + O(5^20))*a^2 - (43015855277311 + O(5^20))*a - 10618306124786 + O(5^20)q^{7} + -(18371543705426 + O(5^20))*a^2 + (12405291764338 + O(5^20))*a + 9307391057342 + O(5^20)q^{8} + -(588118752198*5 + O(5^20))*a^2 - (42988300097032 + O(5^20))*a - 47451623568776 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 18371543705426 + O(5^20)x^{2} + -12405291764594 + O(5^20)x + -9307391057342 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^113q^{2} + z^86q^{3} + z^87q^{4} + z^75q^{6} + z^49q^{7} + z^50q^{8} + z^120q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^51q^{2} + z^24q^{3} + z^87q^{4} + z^19q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^105)\omega^{1}&0\\0&u(z^19)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (36918740115879 + O(5^20))*a^2 + (26995823805512 + O(5^20))*a + 11766912184049 + O(5^20)q^{3} + a^2 - 128q^{4} + -(284390270399*5^2 + O(5^20))*a^2 - (5207406561004*5 + O(5^20))*a + 6598790002149*5 + O(5^20)q^{5} + -(45914080300817 + O(5^20))*a^2 + (602744931637*5^2 + O(5^20))*a + 25133829749243 + O(5^20)q^{6} + (717142301933*5^2 + O(5^20))*a^2 - (43015855277311 + O(5^20))*a - 10618306124786 + O(5^20)q^{7} + -(18371543705426 + O(5^20))*a^2 + (12405291764338 + O(5^20))*a + 9307391057342 + O(5^20)q^{8} + -(588118752198*5 + O(5^20))*a^2 - (42988300097032 + O(5^20))*a - 47451623568776 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 18371543705426 + O(5^20)x^{2} + -12405291764594 + O(5^20)x + -9307391057342 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + q^{3} + 4q^{4} + 4q^{6} + 4q^{7} + 3q^{8} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 2q^{3} + q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -12031215643341 + O(5^20)q^{2} + -31762337230194 + O(5^20)q^{3} + 5809705141769 + O(5^20)q^{4} + 2707153813359*5 + O(5^20)q^{5} + -26731482886846 + O(5^20)q^{6} + 6827855005699 + O(5^20)q^{7} + -23928464154012 + O(5^20)q^{8} + -47618770367047 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 4q^{4} + 4q^{7} + 3q^{8} + 2q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 4q^{2} + 4q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -24256877065986 + O(5^20)q^{2} + -2340702392*5^3 + O(5^17)q^{3} + 41474723932934 + O(5^20)q^{4} + 10665633938*5 + O(5^16)q^{5} + 2590788512*5^3 + O(5^17)q^{6} + 998184332879 + O(5^18)q^{7} + 7198157451908 + O(5^20)q^{8} + -1152493550933 + O(5^18)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^38q^{2} + z^89q^{3} + zq^{4} + z^3q^{6} + z^90q^{7} + z^33q^{8} + z^42q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + z^120q^{3} + z^63q^{4} + z^95q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{0}&0\\0&u(z^95)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (252585850582 + O(5^17))*a^2 + (364359119143 + O(5^17))*a + 263690622298 + O(5^17)q^{3} + a^2 - 512q^{4} + (12487674137*5 + O(5^16))*a^2 - (5855982743*5 + O(5^16))*a + 6573414564*5 + O(5^16)q^{5} + -(168574853363 + O(5^17))*a^2 + (25886343752*5 + O(5^17))*a - 372201466412 + O(5^17)q^{6} + (183939945599*5 + O(5^18))*a^2 - (1771660233631 + O(5^18))*a + 1386595429422 + O(5^18)q^{7} + -(7799436247558 + O(5^20))*a^2 - (27850795755433 + O(5^20))*a - 44768521347816 + O(5^20)q^{8} + (49342961971*5^2 + O(5^18))*a^2 - (1736275877071 + O(5^18))*a + 448626612261 + O(5^18)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 7799436247558 + O(5^20)x^{2} + 27850795754409 + O(5^20)x + 44768521347816 + O(5^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^7q^{2} + z^120q^{3} + z^63q^{4} + z^95q^{5} + z^3q^{6} + z^59q^{7} + z^64q^{8} + z^104q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^35q^{2} + z^104q^{3} + z^67q^{4} + z^103q^{5} + z^15q^{6} + z^47q^{7} + z^72q^{8} + z^24q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^51q^{2} + z^24q^{3} + z^87q^{4} + z^19q^{5} + z^75q^{6} + z^111q^{7} + z^112q^{8} + z^120q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + q^{5} + 4q^{6} + 2q^{7} + q^{8} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^14q^{2} + zq^{3} + z^7q^{4} + 4q^{5} + z^15q^{6} + z^5q^{7} + z^13q^{8} + zq^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^21q^{2} + z^11q^{3} + q^{4} + q^{5} + z^8q^{6} + z^23q^{7} + z^9q^{8} + z^2q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^22q^{2} + z^5q^{3} + z^11q^{4} + 4q^{5} + z^3q^{6} + zq^{7} + z^17q^{8} + z^5q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^9q^{2} + z^7q^{3} + q^{4} + q^{5} + z^16q^{6} + z^19q^{7} + z^21q^{8} + z^10q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 4q^{4} + 3q^{5} + 4q^{7} + 3q^{8} + 2q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + 4q^{4} + 4q^{5} + 2q^{6} + 3q^{7} + 2q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(17);\overline{\mathbf{F}}_{5}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 18, \quad \ell = 5}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + 3q^{2} + 4q^{4} + q^{7} + 2q^{8} + \cdots \in S_{8}(\Gamma_0(18);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 4q^{4} + 4q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 8q^{2} + 64q^{4} + 42*5q^{5} + 1016q^{7} + 512q^{8} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + q^{4} + 3q^{7} + 4q^{8} + \cdots \in S_{10}(\Gamma_0(18);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 4q^{4} + 4q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{0}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -16q^{2} + 256q^{4} + -174*5q^{5} + -952q^{7} + -4096q^{8} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 4q^{4} + 4q^{5} + 4q^{7} + 3q^{8} + \cdots \in S_{8}(\Gamma_0(18);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{4} + 4q^{5} + 2q^{7} + q^{8} + \cdots \in S_{10}(\Gamma_0(18);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + q^{4} + q^{5} + 2q^{7} + 4q^{8} + \cdots \in S_{10}(\Gamma_0(18);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + \cdots \in S_{10}(\Gamma_0(18);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 19, \quad \ell = 5}$ \quad (22 forms)}

\begin{enumerate}
\item  Consider
$f = q + 4q^{3} + 2q^{4} + 3q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 3q^{3} + 3q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{2}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 8701404509683*5 + O(5^20)q^{2} + 21790466780719 + O(5^20)q^{3} + 1446097137097 + O(5^21)q^{4} + 606323604979*5 + O(5^20)q^{5} + -2967279922923*5 + O(5^20)q^{6} + -16194637143797 + O(5^20)q^{7} + -1378813791673*5 + O(5^20)q^{8} + 46146593045399 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + q^{4} + 4q^{7} + 4q^{8} + 3q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -30170798838292 + O(5^20)q^{2} + 1763344147083*5 + O(5^20)q^{3} + -1849572413489 + O(5^20)q^{4} + -7174251360219*5 + O(5^20)q^{5} + 2193548794639*5 + O(5^20)q^{6} + 36789556652224 + O(5^20)q^{7} + 23408701900289 + O(5^20)q^{8} + 24998366860663 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + q^{7} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{2}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -47142929741184 + O(5^20)q^{2} + 45484846421867 + O(5^20)q^{3} + 15643243299853 + O(5^20)q^{4} + -7960661195137*5 + O(5^20)q^{5} + 23573663101347 + O(5^20)q^{6} + -24029740261109 + O(5^20)q^{7} + -1619551238101*5^2 + O(5^20)q^{8} + 40947535372877 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^111q^{2} + z^97q^{3} + z^112q^{4} + z^84q^{6} + z^32q^{7} + z^55q^{8} + z^81q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^49q^{2} + z^35q^{3} + z^112q^{4} + z^13q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^111)\omega^{1}&0\\0&u(z^13)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (42870822084872 + O(5^20))*a^2 - (10478265673048 + O(5^20))*a + 764099362189*5^2 + O(5^20)q^{3} + a^2 - 128q^{4} + -(129547771276*5^2 + O(5^20))*a^2 - (3199196608842*5 + O(5^20))*a - 173638668846*5^2 + O(5^20)q^{5} + -(44132226891159 + O(5^20))*a^2 + (1754163133361 + O(5^20))*a - 15147680600994 + O(5^20)q^{6} + (30452230833798 + O(5^20))*a^2 + (47134683199706 + O(5^20))*a - 731488242534 + O(5^20)q^{7} + -(16061936479463 + O(5^20))*a^2 - (42699945597218 + O(5^20))*a + 5126210581248 + O(5^20)q^{8} + (85707063004523 + O(5^21))*a^2 + (17795599498521*5 + O(5^21))*a - 23603788717304 + O(5^21)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 16061936479463 + O(5^20)x^{2} + 42699945596962 + O(5^20)x + -5126210581248 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^47q^{2} + z^69q^{3} + z^72q^{4} + z^116q^{6} + z^56q^{7} + z^11q^{8} + z^41q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^109q^{2} + z^7q^{3} + z^72q^{4} + z^77q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^47)\omega^{1}&0\\0&u(z^77)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (42870822084872 + O(5^20))*a^2 - (10478265673048 + O(5^20))*a + 764099362189*5^2 + O(5^20)q^{3} + a^2 - 128q^{4} + -(129547771276*5^2 + O(5^20))*a^2 - (3199196608842*5 + O(5^20))*a - 173638668846*5^2 + O(5^20)q^{5} + -(44132226891159 + O(5^20))*a^2 + (1754163133361 + O(5^20))*a - 15147680600994 + O(5^20)q^{6} + (30452230833798 + O(5^20))*a^2 + (47134683199706 + O(5^20))*a - 731488242534 + O(5^20)q^{7} + -(16061936479463 + O(5^20))*a^2 - (42699945597218 + O(5^20))*a + 5126210581248 + O(5^20)q^{8} + (85707063004523 + O(5^21))*a^2 + (17795599498521*5 + O(5^21))*a - 23603788717304 + O(5^21)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 16061936479463 + O(5^20)x^{2} + 42699945596962 + O(5^20)x + -5126210581248 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^59q^{2} + z^113q^{3} + z^64q^{4} + z^48q^{6} + z^36q^{7} + z^27q^{8} + z^33q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^121q^{2} + z^51q^{3} + z^64q^{4} + z^65q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^59)\omega^{1}&0\\0&u(z^65)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (42870822084872 + O(5^20))*a^2 - (10478265673048 + O(5^20))*a + 764099362189*5^2 + O(5^20)q^{3} + a^2 - 128q^{4} + -(129547771276*5^2 + O(5^20))*a^2 - (3199196608842*5 + O(5^20))*a - 173638668846*5^2 + O(5^20)q^{5} + -(44132226891159 + O(5^20))*a^2 + (1754163133361 + O(5^20))*a - 15147680600994 + O(5^20)q^{6} + (30452230833798 + O(5^20))*a^2 + (47134683199706 + O(5^20))*a - 731488242534 + O(5^20)q^{7} + -(16061936479463 + O(5^20))*a^2 - (42699945597218 + O(5^20))*a + 5126210581248 + O(5^20)q^{8} + (85707063004523 + O(5^21))*a^2 + (17795599498521*5 + O(5^21))*a - 23603788717304 + O(5^21)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 16061936479463 + O(5^20)x^{2} + 42699945596962 + O(5^20)x + -5126210581248 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^10080q^{2} + z^504q^{3} + z^6300q^{4} + z^10584q^{6} + z^126q^{7} + z^10836q^{8} + z^2394q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 4x^{3} + x^{2} + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^49q^{2} + z^35q^{3} + z^112q^{4} + z^13q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^111)\omega^{0}&0\\0&u(z^13)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (42776768664626 + O(5^20))*a^2 + (20533639891373 + O(5^20))*a - 982583612877*5^2 + O(5^20)q^{3} + a^2 - 512q^{4} + -(3627661505853*5 + O(5^20))*a^2 + (2830407649658*5 + O(5^20))*a - 1132183846419*5 + O(5^20)q^{5} + (41570386338084 + O(5^20))*a^2 - (25933811061843 + O(5^20))*a - 1502564554204 + O(5^20)q^{6} + -(26603483204304 + O(5^20))*a^2 + (8112460822211 + O(5^20))*a - 8429453082652 + O(5^20)q^{7} + (15420721960836 + O(5^20))*a^2 + (10021068884808 + O(5^20))*a + 32385995966796 + O(5^20)q^{8} + (28780724881538 + O(5^20))*a^2 + (750332938969*5 + O(5^20))*a - 29427738551286 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -15420721960836 + O(5^20)x^{2} + -10021068885832 + O(5^20)x + -32385995966796 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^2016q^{2} + z^12600q^{3} + z^1260q^{4} + z^14616q^{6} + z^3150q^{7} + z^5292q^{8} + z^12978q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 4x^{3} + x^{2} + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^109q^{2} + z^7q^{3} + z^72q^{4} + z^77q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^47)\omega^{0}&0\\0&u(z^77)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (42776768664626 + O(5^20))*a^2 + (20533639891373 + O(5^20))*a - 982583612877*5^2 + O(5^20)q^{3} + a^2 - 512q^{4} + -(3627661505853*5 + O(5^20))*a^2 + (2830407649658*5 + O(5^20))*a - 1132183846419*5 + O(5^20)q^{5} + (41570386338084 + O(5^20))*a^2 - (25933811061843 + O(5^20))*a - 1502564554204 + O(5^20)q^{6} + -(26603483204304 + O(5^20))*a^2 + (8112460822211 + O(5^20))*a - 8429453082652 + O(5^20)q^{7} + (15420721960836 + O(5^20))*a^2 + (10021068884808 + O(5^20))*a + 32385995966796 + O(5^20)q^{8} + (28780724881538 + O(5^20))*a^2 + (750332938969*5 + O(5^20))*a - 29427738551286 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -15420721960836 + O(5^20)x^{2} + -10021068885832 + O(5^20)x + -32385995966796 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^3528q^{2} + z^2520q^{3} + z^252q^{4} + z^6048q^{6} + z^630q^{7} + z^7308q^{8} + z^11970q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 4x^{3} + x^{2} + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^121q^{2} + z^51q^{3} + z^64q^{4} + z^65q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^59)\omega^{0}&0\\0&u(z^65)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (42776768664626 + O(5^20))*a^2 + (20533639891373 + O(5^20))*a - 982583612877*5^2 + O(5^20)q^{3} + a^2 - 512q^{4} + -(3627661505853*5 + O(5^20))*a^2 + (2830407649658*5 + O(5^20))*a - 1132183846419*5 + O(5^20)q^{5} + (41570386338084 + O(5^20))*a^2 - (25933811061843 + O(5^20))*a - 1502564554204 + O(5^20)q^{6} + -(26603483204304 + O(5^20))*a^2 + (8112460822211 + O(5^20))*a - 8429453082652 + O(5^20)q^{7} + (15420721960836 + O(5^20))*a^2 + (10021068884808 + O(5^20))*a + 32385995966796 + O(5^20)q^{8} + (28780724881538 + O(5^20))*a^2 + (750332938969*5 + O(5^20))*a - 29427738551286 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -15420721960836 + O(5^20)x^{2} + -10021068885832 + O(5^20)x + -32385995966796 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{3} + 3q^{4} + 4q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 3q^{3} + 3q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 5576470313142*5 + O(5^20)q^{2} + 44783519563928 + O(5^20)q^{3} + -15230936962037 + O(5^21)q^{4} + 405632409603*5 + O(5^19)q^{5} + 1743580354276*5 + O(5^20)q^{6} + -41370428642036 + O(5^20)q^{7} + 4226288012292*5 + O(5^20)q^{8} + 22338317832001 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 4q^{4} + 2q^{7} + 3q^{8} + 2q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 47507616025214 + O(5^20)q^{2} + 1393955669912*5 + O(5^19)q^{3} + 38285381401534 + O(5^20)q^{4} + 1274870049527*5 + O(5^19)q^{5} + -1826165588832*5 + O(5^19)q^{6} + -3248556141933 + O(5^19)q^{7} + 7983068462458 + O(5^20)q^{8} + -9076979076083 + O(5^19)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 2q^{4} + 2q^{6} + 3q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -10776133410992 + O(5^20)q^{2} + -6416144924401 + O(5^19)q^{3} + 32317509079802 + O(5^20)q^{4} + -1250964593171*5 + O(5^19)q^{5} + -8986182740458 + O(5^19)q^{6} + -2916495708102 + O(5^19)q^{7} + -1808542111136*5 + O(5^20)q^{8} + -1254744810882 + O(5^19)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{4} + 3q^{5} + q^{7} + q^{8} + 3q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^109q^{2} + z^7q^{3} + z^72q^{4} + z^77q^{5} + z^116q^{6} + z^118q^{7} + z^73q^{8} + z^41q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^121q^{2} + z^51q^{3} + z^64q^{4} + z^65q^{5} + z^48q^{6} + z^98q^{7} + z^89q^{8} + z^33q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^49q^{2} + z^35q^{3} + z^112q^{4} + z^13q^{5} + z^84q^{6} + z^94q^{7} + z^117q^{8} + z^81q^{9} + \cdots \in S_{8}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + 2q^{6} + 3q^{7} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 4x^{3} + x^{2} + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^12369q^{2} + z^14322q^{3} + z^8463q^{4} + q^{5} + z^11067q^{6} + z^4557q^{7} + z^14322q^{8} + z^5859q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 4x^{3} + x^{2} + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^14973q^{2} + z^9114q^{3} + z^11067q^{4} + q^{5} + z^8463q^{6} + z^7161q^{7} + z^9114q^{8} + z^13671q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 4x^{3} + x^{2} + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{3} + 3q^{4} + 3q^{5} + 4q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + q^{6} + 3q^{7} + 3q^{8} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^24q^{2} + z^26q^{3} + z^120q^{4} + 4q^{5} + z^50q^{6} + z^122q^{7} + z^67q^{8} + z^53q^{9} + \cdots \in S_{10}(\Gamma_0(19);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{3} + 3x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 21, \quad \ell = 5}$ \quad (13 forms)}

\begin{enumerate}
\item  Consider
$f = q + 3q^{2} + 3q^{3} + q^{4} + 4q^{6} + 3q^{7} + 4q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 4q^{2} + q^{3} + 4q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{2}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -27q^{3} + -9*a + 118q^{4} + -4*5*a - 54*5q^{5} + -27*aq^{6} + 343q^{7} + 71*a - 2214q^{8} + 729q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 9x + -246$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^19q^{2} + 2q^{3} + z^10q^{4} + zq^{6} + 3q^{7} + q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + 3q^{3} + z^10q^{4} + zq^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^23)\omega^{1}&0\\0&u(z)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 27q^{3} + -(6160244277797 + O(5^20))*a - 1300495433396 + O(5^20)q^{4} + (2464097711122*5 + O(5^20))*a + 520198173384*5 + O(5^20)q^{5} + 27*aq^{6} + 343q^{7} + (3696146566687*5 + O(5^20))*a + 3901486300596 + O(5^20)q^{8} + 729q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 6160244277797 + O(5^20)x + 1300495433268 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^23q^{2} + 2q^{3} + z^2q^{4} + z^5q^{6} + 3q^{7} + q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^11q^{2} + 3q^{3} + z^2q^{4} + z^5q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^19)\omega^{1}&0\\0&u(z^5)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 27q^{3} + -(6160244277797 + O(5^20))*a - 1300495433396 + O(5^20)q^{4} + (2464097711122*5 + O(5^20))*a + 520198173384*5 + O(5^20)q^{5} + 27*aq^{6} + 343q^{7} + (3696146566687*5 + O(5^20))*a + 3901486300596 + O(5^20)q^{8} + 729q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 6160244277797 + O(5^20)x + 1300495433268 + O(5^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^13q^{2} + 4q^{3} + z^22q^{4} + zq^{6} + 4q^{7} + 2q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + 3q^{3} + z^10q^{4} + zq^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^23)\omega^{0}&0\\0&u(z)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -81q^{3} + 9*a + 56q^{4} + -14*5*a + 36*5^2q^{5} + -81*aq^{6} + -2401q^{7} + -3*5^3*a + 5112q^{8} + 6561q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -9x + -568$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^17q^{2} + 4q^{3} + z^14q^{4} + z^5q^{6} + 4q^{7} + 2q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^11q^{2} + 3q^{3} + z^2q^{4} + z^5q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^19)\omega^{0}&0\\0&u(z^5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -81q^{3} + 9*a + 56q^{4} + -14*5*a + 36*5^2q^{5} + -81*aq^{6} + -2401q^{7} + -3*5^3*a + 5112q^{8} + 6561q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -9x + -568$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + q^{3} + 4q^{4} + 4q^{6} + 4q^{7} + 3q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 4q^{2} + q^{3} + 4q^{4} + 3q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(3)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 81q^{3} + (6667289243573 + O(5^20))*a - 26029455799893 + O(5^20)q^{4} + -(533383139484*5 + O(5^19))*a - 1732340801421*5 + O(5^19)q^{5} + 81*aq^{6} + -2401q^{7} + (8692671474674 + O(5^20))*a - 43086801159563 + O(5^20)q^{8} + 6561q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -6667289243573 + O(5^20)x + 26029455799381 + O(5^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 2q^{3} + q^{4} + 2q^{5} + 4q^{6} + 2q^{7} + q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^7q^{2} + 3q^{3} + z^10q^{4} + zq^{5} + zq^{6} + 2q^{7} + 4q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^11q^{2} + 3q^{3} + z^2q^{4} + z^5q^{5} + z^5q^{6} + 2q^{7} + 4q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 2q^{3} + 3q^{4} + q^{5} + 3q^{6} + 3q^{7} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + 4q^{4} + q^{5} + q^{6} + q^{7} + 2q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{3} + 3q^{4} + 4q^{5} + q^{7} + q^{9} + \cdots \in S_{10}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + q^{3} + 4q^{4} + 3q^{5} + 4q^{6} + 4q^{7} + 3q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(21);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 22, \quad \ell = 5}$ \quad (12 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + q^{3} + 4q^{4} + 2q^{6} + 3q^{7} + 3q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -8q^{2} + 91q^{3} + 64q^{4} + 37*5q^{5} + -728q^{6} + -722q^{7} + -512q^{8} + 6094q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 4q^{4} + 2q^{6} + 2q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + 4q^{4} + 2q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(2)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 8q^{2} + -7360483388691 + O(5^20)q^{3} + 64q^{4} + 1472096677769*5 + O(5^20)q^{5} + 36483564531097 + O(5^20)q^{6} + 1535867160357*5 + O(5^20)q^{7} + 512q^{8} + -21443745339956 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + 4q^{7} + q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{0}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 16q^{2} + 137q^{3} + 256q^{4} + -119*5q^{5} + 2192q^{6} + 11354q^{7} + 4096q^{8} + -914q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 3q^{3} + q^{4} + 2q^{6} + 4q^{8} + q^{9} + \cdots \in S_{10}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + q^{3} + 4q^{4} + 2q^{5} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -16q^{2} + 28098984650238 + O(5^20)q^{3} + 256q^{4} + -9367194068197*5 + O(5^20)q^{5} + 27253403799317 + O(5^20)q^{6} + 3064871497298*5^2 + O(5^21)q^{7} + -4096q^{8} + -17874087813039 + O(5^20)q^{9} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{3} + 4q^{4} + 2q^{5} + 2q^{6} + 3q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + 2q^{8} + 4q^{9} + \cdots \in S_{8}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 3q^{3} + 4q^{4} + q^{5} + 4q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + \cdots \in S_{8}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 4q^{3} + q^{4} + q^{5} + 4q^{6} + 3q^{7} + q^{8} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + 4q^{5} + q^{6} + 4q^{7} + q^{8} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + q^{3} + q^{4} + 4q^{5} + 4q^{6} + 4q^{8} + 3q^{9} + \cdots \in S_{10}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + z^13q^{3} + q^{4} + q^{5} + zq^{6} + zq^{7} + 4q^{8} + zq^{9} + \cdots \in S_{10}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + z^17q^{3} + q^{4} + q^{5} + z^5q^{6} + z^5q^{7} + 4q^{8} + z^5q^{9} + \cdots \in S_{10}(\Gamma_0(22);\overline{\mathbf{F}}_{5}).$
Also,~$z$ satisfies the equation $x^{2} + 4x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\setcounter{section}{6}
\section{$\mathbf{2 \leq N \leq 16, \quad \ell = 7}$}

\subsection{$\mathbf{N = 2, \quad \ell = 7}$ \quad (3 forms)}

\begin{enumerate}
\item[-]
----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 5q^{3} + 4q^{4} + 2q^{5} + 3q^{6} + q^{8} + 5q^{9} + 4q^{10} + 6q^{11} + 6q^{12} + \cdots \in S_{10}(\Gamma_0(2);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 4q^{3} + q^{4} + 6q^{5} + 4q^{6} + q^{8} + 6q^{9} + 6q^{10} + 5q^{11} + 4q^{12} + \cdots \in S_{14}(\Gamma_0(2);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 6q^{2} + 5q^{3} + q^{4} + 2q^{6} + q^{7} + 6q^{8} + q^{9} + 5q^{12} + 3q^{13} + \cdots \in S_{14}(\Gamma_0(2);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 3, \quad \ell = 7}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + 4q^{2} + 4q^{3} + q^{4} + 3q^{5} + 2q^{6} + 2q^{9} + 5q^{10} + 6q^{11} + 4q^{12} + q^{13} + \cdots \in S_{10}(\Gamma_0(3);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{2} + 2q^{3} + 4q^{4} + 6q^{5} + 2q^{6} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(2)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 18q^{2} + 81q^{3} + -188q^{4} + -1530q^{5} + 1458q^{6} + 1304*7q^{7} + -1800*7q^{8} + 6561q^{9} + -27540q^{10} + 21132q^{11} + -15228q^{12} + 31214q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 6q^{3} + 2q^{4} + 2q^{5} + 5q^{6} + q^{9} + 4q^{10} + 5q^{11} + 5q^{12} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(3);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{2} + 2q^{3} + 4q^{4} + 6q^{5} + 2q^{6} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -12q^{2} + -729q^{3} + -8048q^{4} + -30210q^{5} + 8748q^{6} + 33584*7q^{7} + 27840*7q^{8} + 531441q^{9} + 362520q^{10} + -11182908q^{11} + 5866992q^{12} + 8049614q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 6q^{2} + 3q^{3} + 2q^{5} + 4q^{6} + q^{8} + 2q^{9} + 5q^{10} + 6q^{11} + \cdots \in S_{10}(\Gamma_0(3);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 2q^{3} + 4q^{4} + 6q^{5} + 2q^{6} + 2q^{7} + 4q^{9} + 6q^{10} + 3q^{11} + q^{12} + q^{13} + \cdots \in S_{12}(\Gamma_0(3);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + q^{3} + 6q^{4} + 5q^{5} + 6q^{6} + 6q^{7} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + 6q^{12} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(3);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + q^{3} + 6q^{5} + 3q^{6} + q^{8} + q^{9} + 4q^{10} + 5q^{11} + \cdots \in S_{14}(\Gamma_0(3);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 4, \quad \ell = 7}$ \quad (3 forms)}

\begin{enumerate}
\item  Consider
$f = q + 4q^{3} + 6q^{5} + 3q^{9} + 2q^{11} + 2q^{13} + \cdots \in S_{10}(\Gamma_0(4);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{3} + 5q^{5} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 228q^{3} + -666q^{5} + -904*7q^{7} + 32301q^{9} + -30420q^{11} + -32338q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{3} + 4q^{5} + 5q^{9} + 4q^{11} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(4);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{3} + 5q^{5} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 468q^{3} + 56214q^{5} + 47576*7q^{7} + -1375299q^{9} + -6397380q^{11} + 15199742q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{3} + 5q^{5} + 3q^{7} + 6q^{9} + q^{11} + 2q^{13} + \cdots \in S_{12}(\Gamma_0(4);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 5, \quad \ell = 7}$ \quad (11 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 6q^{3} + 2q^{5} + 6q^{6} + 6q^{8} + 2q^{9} + 2q^{10} + 5q^{11} + 6q^{13} + \cdots \in S_{10}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 3q^{3} + 4q^{5} + 6q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -36436499247640568 + O(7^20)q^{2} + -6919267802330745 + O(7^20)q^{3} + -4942334144521979*7 + O(7^20)q^{4} + 625q^{5} + -74197117920125425 + O(7^21)q^{6} + -3175894024413538*7 + O(7^20)q^{7} + -31798764822919893 + O(7^20)q^{8} + 36212496239049776 + O(7^20)q^{9} + -32016134955934715 + O(7^20)q^{10} + 30338575810159160 + O(7^20)q^{11} + 4204185432103806*7 + O(7^20)q^{12} + 30318677183494069 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 6q^{3} + 4q^{4} + 3q^{5} + q^{6} + 3q^{9} + 4q^{10} + 2q^{11} + 3q^{12} + 3q^{13} + \cdots \in S_{12}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 2q^{3} + q^{4} + 2q^{5} + 6q^{6} + 6q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{4}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 34q^{2} + -792q^{3} + -892q^{4} + 3125q^{5} + -26928q^{6} + -2508*7q^{7} + -2040*7^2q^{8} + 450117q^{9} + 106250q^{10} + -468788q^{11} + 706464q^{12} + -374042q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + q^{3} + 4q^{4} + 4q^{5} + 6q^{6} + 3q^{9} + 3q^{10} + 2q^{11} + 4q^{12} + 4q^{13} + \cdots \in S_{12}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 2q^{3} + q^{4} + 2q^{5} + 6q^{6} + 6q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 22817967818630025 + O(7^20)q^{2} + 15306139969210742 + O(7^20)q^{3} + 22394241413074794 + O(7^20)q^{4} + -3125q^{5} + 1486159987734441774 + O(7^22)q^{6} + 3764145885473580*7 + O(7^20)q^{7} + -157953003842027*7 + O(7^20)q^{8} + -16075608726693821 + O(7^20)q^{9} + 28136636846300769 + O(7^20)q^{10} + 39017467144789429 + O(7^20)q^{11} + 16983759292187958 + O(7^20)q^{12} + -18127525411569724 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 2q^{3} + 6q^{5} + q^{6} + 6q^{8} + q^{9} + 3q^{10} + 3q^{11} + q^{13} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 3q^{3} + 4q^{5} + 6q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 33691825955258356 + O(7^20)q^{2} + -5340579975040357 + O(7^20)q^{3} + 358999544859954*7^2 + O(7^20)q^{4} + -15625q^{5} + 28126849851088745 + O(7^20)q^{6} + -2375535549637855*7 + O(7^20)q^{7} + -6328666560350800 + O(7^20)q^{8} + -16454533056251135 + O(7^20)q^{9} + 34592480732170098 + O(7^20)q^{10} + 23623575968533075 + O(7^20)q^{11} + -307350181470413*7^2 + O(7^20)q^{12} + -1444793968139479 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 6q^{2} + 5q^{3} + 5q^{5} + 2q^{6} + q^{8} + 5q^{9} + 2q^{10} + 6q^{11} + \cdots \in S_{10}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 3q^{2} + 2q^{3} + q^{4} + 2q^{5} + 6q^{6} + 6q^{7} + 5q^{9} + 6q^{10} + 4q^{11} + 2q^{12} + 4q^{13} + \cdots \in S_{10}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 3q^{3} + 4q^{5} + 6q^{6} + 3q^{7} + 6q^{8} + 4q^{9} + q^{10} + 6q^{11} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 5q^{4} + 6q^{5} + q^{7} + 5q^{9} + 4q^{11} + 5q^{12} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^25q^{2} + z^31q^{3} + z^38q^{4} + q^{5} + 3q^{6} + 6q^{7} + z^26q^{8} + z^35q^{9} + z^25q^{10} + z^7q^{11} + z^21q^{12} + z^26q^{13} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^31q^{2} + z^25q^{3} + z^26q^{4} + q^{5} + 3q^{6} + 6q^{7} + z^38q^{8} + z^5q^{9} + z^31q^{10} + zq^{11} + z^3q^{12} + z^38q^{13} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + q^{5} + 5q^{6} + q^{8} + 6q^{9} + 3q^{10} + 5q^{11} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 6, \quad \ell = 7}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + 3q^{2} + 2q^{3} + 2q^{4} + 5q^{5} + 6q^{6} + 6q^{8} + 4q^{9} + q^{10} + 6q^{11} + 4q^{12} + 3q^{13} + \cdots \in S_{12}(\Gamma_0(6);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 5q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 5q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(5)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -32q^{2} + -243q^{3} + 1024q^{4} + 5766q^{5} + 7776q^{6} + 10352*7q^{7} + -32768q^{8} + 59049q^{9} + -184512q^{10} + -408948q^{11} + -248832q^{12} + 1367558q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 5q^{3} + 2q^{4} + 2q^{5} + q^{6} + 6q^{8} + 4q^{9} + 6q^{10} + 6q^{11} + 3q^{12} + 4q^{13} + \cdots \in S_{12}(\Gamma_0(6);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 5q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 5q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{4}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -32q^{2} + 243q^{3} + 1024q^{4} + -11730q^{5} + -7776q^{6} + -7144*7q^{7} + -32768q^{8} + 59049q^{9} + 375360q^{10} + -531420q^{11} + 248832q^{12} + 1332566q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 5q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 5q^{7} + 6q^{8} + 2q^{9} + 2q^{10} + 5q^{11} + 2q^{12} + 3q^{13} + \cdots \in S_{10}(\Gamma_0(6);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 5q^{3} + 2q^{4} + 4q^{5} + 6q^{6} + q^{7} + q^{8} + 4q^{9} + 2q^{10} + 3q^{11} + 3q^{12} + \cdots \in S_{12}(\Gamma_0(6);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 6q^{3} + q^{4} + 5q^{5} + 6q^{6} + 6q^{7} + q^{8} + q^{9} + 5q^{10} + 3q^{11} + 6q^{12} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(6);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 8, \quad \ell = 7}$ \quad (8 forms)}

\begin{enumerate}
\item  Consider
$f = q + 5q^{3} + 5q^{5} + 5q^{9} + 3q^{11} + 2q^{13} + \cdots \in S_{10}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 6q^{3} + 3q^{5} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 68q^{3} + 1510q^{5} + 1464*7q^{7} + -15059q^{9} + 3916q^{11} + -176594q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{3} + 3q^{5} + 6q^{9} + 6q^{11} + q^{13} + \cdots \in S_{12}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 5q^{5} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -26125321683010496 + O(7^20)q^{3} + -5665204994317782 + O(7^20)q^{5} + 2832602497167362*7 + O(7^21)q^{7} + 212619578001532778 + O(7^21)q^{9} + 231380275661284448 + O(7^21)q^{11} + -31202403233731204 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{3} + 4q^{5} + 6q^{9} + 6q^{11} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 5q^{5} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{4}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 26125321683010552 + O(7^20)q^{3} + 5665204994325650 + O(7^20)q^{5} + -2832602497154354*7 + O(7^21)q^{7} + -212619578000992576 + O(7^21)q^{9} + -231380275661125368 + O(7^21)q^{11} + 31202403234781680 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{3} + q^{5} + 6q^{9} + 6q^{11} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 6q^{3} + 3q^{5} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 19813083833991809 + O(7^20)q^{3} + -1619792884930289 + O(7^20)q^{5} + -1418566396767041*7 + O(7^20)q^{7} + -37912683339532212 + O(7^20)q^{9} + -11831635427128049 + O(7^20)q^{11} + -25421190565315715 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{3} + 5q^{5} + 3q^{7} + 3q^{9} + 5q^{11} + q^{13} + \cdots \in S_{10}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{3} + 3q^{5} + 4q^{7} + 3q^{9} + 5q^{11} + 2q^{13} + \cdots \in S_{12}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + 3q^{5} + q^{7} + q^{9} + \cdots \in S_{14}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{5} + 6q^{7} + 4q^{9} + 3q^{11} + 2q^{13} + \cdots \in S_{14}(\Gamma_0(8);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 9, \quad \ell = 7}$ \quad (11 forms)}

\begin{enumerate}
\item  Consider
$f = q + 3q^{2} + q^{4} + 4q^{5} + 5q^{10} + q^{11} + q^{13} + \cdots \in S_{10}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 6q^{2} + 4q^{4} + q^{5} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(2)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -18q^{2} + -188q^{4} + 1530q^{5} + 1304*7q^{7} + 1800*7q^{8} + -27540q^{10} + -21132q^{11} + 31214q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{4} + \cdots \in S_{12}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 6q^{4} + 6q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{2}&0\\0&u(6)\omega^{5}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 472q^{4} + 32*7*aq^{5} + 8300*7q^{7} + -1576*aq^{8} + 11520*7^2q^{10} + -3200*aq^{11} + 108950*7q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -360*7$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 2q^{4} + 5q^{5} + 4q^{10} + 2q^{11} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 6q^{2} + 4q^{4} + q^{5} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 12q^{2} + -8048q^{4} + 30210q^{5} + 33584*7q^{7} + -27840*7q^{8} + 362520q^{10} + 11182908q^{11} + 8049614q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 6q^{4} + 6q^{7} + \cdots \in S_{10}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 5q^{5} + 6q^{8} + 5q^{10} + q^{11} + \cdots \in S_{10}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 3q^{2} + 5q^{4} + 3q^{8} + 6q^{11} + \cdots \in S_{12}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 6q^{2} + 4q^{4} + q^{5} + 2q^{7} + 6q^{10} + 4q^{11} + q^{13} + \cdots \in S_{12}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^28q^{2} + q^{4} + z^20q^{5} + q^{7} + z^4q^{8} + q^{10} + z^44q^{11} + 2q^{13} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^4q^{2} + q^{4} + z^44q^{5} + q^{7} + z^28q^{8} + q^{10} + z^20q^{11} + 2q^{13} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + q^{5} + 6q^{8} + 4q^{10} + 2q^{11} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 6q^{4} + 2q^{5} + 6q^{7} + 4q^{8} + 2q^{10} + 3q^{11} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 10, \quad \ell = 7}$ \quad (11 forms)}

\begin{enumerate}
\item  Consider
$f = q + 5q^{2} + 4q^{3} + 4q^{4} + 5q^{5} + 6q^{6} + 6q^{8} + 3q^{9} + 4q^{10} + 4q^{11} + 2q^{12} + 3q^{13} + \cdots \in S_{10}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(6)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -16q^{2} + 46q^{3} + 256q^{4} + -625q^{5} + -736q^{6} + -1474*7q^{7} + -4096q^{8} + -17567q^{9} + 10000q^{10} + -5568q^{11} + 11776q^{12} + 45986q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 6q^{3} + 4q^{4} + 2q^{5} + 2q^{6} + 6q^{8} + 2q^{9} + 3q^{10} + 5q^{11} + 3q^{12} + 6q^{13} + \cdots \in S_{10}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 3q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -16q^{2} + -204q^{3} + 256q^{4} + 625q^{5} + 3264q^{6} + 776*7q^{7} + -4096q^{8} + 21933q^{9} + -10000q^{10} + 73932q^{11} + -52224q^{12} + -114514q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 4q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + q^{8} + 4q^{9} + 2q^{10} + 6q^{11} + q^{12} + 2q^{13} + \cdots \in S_{12}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 6q^{3} + 4q^{4} + 5q^{5} + 5q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{4}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 32q^{2} + -318q^{3} + 1024q^{4} + -3125q^{5} + -10176q^{6} + -10102*7q^{7} + 32768q^{8} + -76023q^{9} + -100000q^{10} + 238272q^{11} + -325632q^{12} + -2097478q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 3q^{3} + 2q^{4} + 3q^{5} + 5q^{6} + q^{8} + 4q^{9} + 5q^{10} + 6q^{11} + 6q^{12} + 5q^{13} + \cdots \in S_{12}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 6q^{3} + 4q^{4} + 5q^{5} + 5q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 32q^{2} + -16199033841643874 + O(7^20)q^{3} + 1024q^{4} + 3125q^{5} + -39615485146931962 + O(7^20)q^{6} + -756085249000834*7 + O(7^20)q^{7} + 32768q^{8} + 30232314253304776 + O(7^20)q^{9} + 100000q^{10} + -33439468469052759 + O(7^20)q^{11} + 8980736059969232 + O(7^20)q^{12} + 17721419729062005 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 2q^{3} + q^{4} + 6q^{5} + 5q^{6} + 6q^{8} + q^{9} + q^{10} + 3q^{11} + 2q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 3q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -64q^{2} + -26q^{3} + 4096q^{4} + -15625q^{5} + 1664q^{6} + 76934*7q^{7} + -262144q^{8} + -1593647q^{9} + 1000000q^{10} + -3994848q^{11} + -106496q^{12} + -23834446q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 6q^{3} + q^{4} + q^{5} + q^{6} + 6q^{8} + 5q^{9} + 6q^{10} + q^{11} + 6q^{12} + 4q^{13} + \cdots \in S_{14}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{0}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -64q^{2} + 1224q^{3} + 4096q^{4} + 15625q^{5} + -78336q^{6} + -9316*7q^{7} + -262144q^{8} + -96147q^{9} + -1000000q^{10} + 7427652q^{11} + 5013504q^{12} + 32243054q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 6q^{3} + 4q^{4} + 5q^{5} + 5q^{6} + 3q^{7} + q^{8} + 2q^{9} + 3q^{10} + 5q^{11} + 3q^{12} + 5q^{13} + \cdots \in S_{10}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + 6q^{8} + 6q^{9} + 2q^{10} + 2q^{11} + 4q^{12} + 3q^{13} + \cdots \in S_{12}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 3q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 3q^{7} + 6q^{8} + 4q^{9} + 5q^{10} + 6q^{11} + 6q^{12} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 6q^{3} + 2q^{4} + 3q^{5} + 3q^{6} + q^{7} + q^{8} + 3q^{9} + 5q^{10} + 3q^{11} + 5q^{12} + \cdots \in S_{12}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{4} + 6q^{5} + 6q^{7} + q^{8} + 4q^{9} + 6q^{10} + 4q^{11} + q^{13} + \cdots \in S_{14}(\Gamma_0(10);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 11, \quad \ell = 7}$ \quad (23 forms)}

\begin{enumerate}
\item  Consider
$f = q + 5q^{2} + 5q^{3} + 3q^{4} + q^{5} + 4q^{6} + 3q^{8} + 5q^{9} + 5q^{10} + 3q^{11} + q^{12} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 6q^{3} + 5q^{4} + 2q^{5} + 4q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -30871450180857876 + O(7^20)q^{2} + 4773729341824567 + O(7^20)q^{3} + -36314553836254201 + O(7^20)q^{4} + -24637304674161458 + O(7^20)q^{5} + -22136466302621933 + O(7^20)q^{6} + -5365660500392004*7 + O(7^20)q^{7} + -30285531255457811 + O(7^20)q^{8} + -2251524041307151 + O(7^20)q^{9} + 35992958622474859 + O(7^20)q^{10} + -14641q^{11} + 235045208610571934 + O(7^21)q^{12} + 390369396082832*7 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + q^{4} + 5q^{5} + 5q^{6} + 3q^{9} + q^{10} + 4q^{11} + 4q^{12} + 3q^{13} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 5q^{2} + 6q^{3} + 2q^{4} + q^{5} + 2q^{6} + 5q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{2}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 24777401154309464 + O(7^20)q^{2} + 9068164705809584 + O(7^20)q^{3} + 5636087969631820 + O(7^20)q^{4} + -11708894028337977 + O(7^20)q^{5} + 13799444035639533 + O(7^20)q^{6} + -38139629513736171*7 + O(7^21)q^{7} + -2086150468467182*7 + O(7^20)q^{8} + -17767431542269455 + O(7^20)q^{9} + -22385172834451748 + O(7^20)q^{10} + 14641q^{11} + -15003190329497842 + O(7^20)q^{12} + 22403546794310594 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^72q^{2} + z^9q^{3} + z^105q^{4} + z^30q^{5} + z^81q^{6} + z^58q^{8} + z^266q^{9} + z^102q^{10} + 4q^{11} + 2q^{12} + z^54q^{13} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^186q^{2} + z^237q^{3} + z^333q^{4} + z^144q^{5} + z^81q^{6} + z^76q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^266)\omega^{1}&0\\0&u(z^76)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (27837678167433989 + O(7^20))*a^2 + (4061054072657776 + O(7^20))*a + 35042109046646096 + O(7^20)q^{3} + a^2 - 512q^{4} + (38034084380105790 + O(7^20))*a^2 - (5149784015446046*7 + O(7^20))*a + 41755180074486*7 + O(7^20)q^{5} + -(39119502853433172 + O(7^20))*a^2 + (4540937842883394 + O(7^20))*a + 7981711151246624 + O(7^20)q^{6} + -(8277926740717399*7 + O(7^21))*a^2 + (33564610154706479*7 + O(7^21))*a - 39032964622270567*7 + O(7^21)q^{7} + -(51222887969695*7 + O(7^20))*a^2 - (24130096886840683 + O(7^20))*a + 37908942984174977 + O(7^20)q^{8} + (18245030635494335 + O(7^20))*a^2 - (24253935591846439 + O(7^20))*a + 28618913636236755 + O(7^20)q^{9} + (30118047187886713*7 + O(7^21))*a^2 + (12374366438588216 + O(7^21))*a - 159431918881716372 + O(7^21)q^{10} + 14641q^{11} + -(5619986227906564*7 + O(7^20))*a^2 + (418007817858361*7 + O(7^20))*a - 38235950949598526 + O(7^20)q^{12} + (12287174735587712 + O(7^20))*a^2 + (26605374927772883 + O(7^20))*a - 36462484578826797 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 51222887969695*7 + O(7^20)x^{2} + 24130096886839659 + O(7^20)x + -37908942984174977 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^108q^{2} + z^99q^{3} + z^15q^{4} + z^102q^{5} + z^207q^{6} + z^106q^{8} + z^38q^{9} + z^210q^{10} + 4q^{11} + 2q^{12} + z^252q^{13} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^222q^{2} + z^327q^{3} + z^243q^{4} + z^216q^{5} + z^207q^{6} + z^304q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^38)\omega^{1}&0\\0&u(z^304)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (27837678167433989 + O(7^20))*a^2 + (4061054072657776 + O(7^20))*a + 35042109046646096 + O(7^20)q^{3} + a^2 - 512q^{4} + (38034084380105790 + O(7^20))*a^2 - (5149784015446046*7 + O(7^20))*a + 41755180074486*7 + O(7^20)q^{5} + -(39119502853433172 + O(7^20))*a^2 + (4540937842883394 + O(7^20))*a + 7981711151246624 + O(7^20)q^{6} + -(8277926740717399*7 + O(7^21))*a^2 + (33564610154706479*7 + O(7^21))*a - 39032964622270567*7 + O(7^21)q^{7} + -(51222887969695*7 + O(7^20))*a^2 - (24130096886840683 + O(7^20))*a + 37908942984174977 + O(7^20)q^{8} + (18245030635494335 + O(7^20))*a^2 - (24253935591846439 + O(7^20))*a + 28618913636236755 + O(7^20)q^{9} + (30118047187886713*7 + O(7^21))*a^2 + (12374366438588216 + O(7^21))*a - 159431918881716372 + O(7^21)q^{10} + 14641q^{11} + -(5619986227906564*7 + O(7^20))*a^2 + (418007817858361*7 + O(7^20))*a - 38235950949598526 + O(7^20)q^{12} + (12287174735587712 + O(7^20))*a^2 + (26605374927772883 + O(7^20))*a - 36462484578826797 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 51222887969695*7 + O(7^20)x^{2} + 24130096886839659 + O(7^20)x + -37908942984174977 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^162q^{2} + z^63q^{3} + z^51q^{4} + z^210q^{5} + z^225q^{6} + z^64q^{8} + z^152q^{9} + z^30q^{10} + 4q^{11} + 2q^{12} + z^36q^{13} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^276q^{2} + z^291q^{3} + z^279q^{4} + z^324q^{5} + z^225q^{6} + z^190q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^152)\omega^{1}&0\\0&u(z^190)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (27837678167433989 + O(7^20))*a^2 + (4061054072657776 + O(7^20))*a + 35042109046646096 + O(7^20)q^{3} + a^2 - 512q^{4} + (38034084380105790 + O(7^20))*a^2 - (5149784015446046*7 + O(7^20))*a + 41755180074486*7 + O(7^20)q^{5} + -(39119502853433172 + O(7^20))*a^2 + (4540937842883394 + O(7^20))*a + 7981711151246624 + O(7^20)q^{6} + -(8277926740717399*7 + O(7^21))*a^2 + (33564610154706479*7 + O(7^21))*a - 39032964622270567*7 + O(7^21)q^{7} + -(51222887969695*7 + O(7^20))*a^2 - (24130096886840683 + O(7^20))*a + 37908942984174977 + O(7^20)q^{8} + (18245030635494335 + O(7^20))*a^2 - (24253935591846439 + O(7^20))*a + 28618913636236755 + O(7^20)q^{9} + (30118047187886713*7 + O(7^21))*a^2 + (12374366438588216 + O(7^21))*a - 159431918881716372 + O(7^21)q^{10} + 14641q^{11} + -(5619986227906564*7 + O(7^20))*a^2 + (418007817858361*7 + O(7^20))*a - 38235950949598526 + O(7^20)q^{12} + (12287174735587712 + O(7^20))*a^2 + (26605374927772883 + O(7^20))*a - 36462484578826797 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 51222887969695*7 + O(7^20)x^{2} + 24130096886839659 + O(7^20)x + -37908942984174977 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^33q^{2} + z^10q^{3} + z^6q^{4} + z^33q^{5} + z^43q^{6} + z^2q^{8} + z^33q^{9} + z^18q^{10} + 5q^{11} + 2q^{12} + z^6q^{13} + \cdots \in S_{12}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^17q^{2} + z^26q^{3} + z^22q^{4} + z^17q^{5} + z^43q^{6} + z^25q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^23)\omega^{1}&0\\0&u(z^25)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (1526861999260261 + O(7^19))*a - 1594789776487798 + O(7^19)q^{3} + -(25739792713100455 + O(7^20))*a - 17835578556245635 + O(7^20)q^{4} + (3427501219660162 + O(7^19))*a - 282490867335*7^5 + O(7^19)q^{5} + -(1930629739857657 + O(7^19))*a + 2692481377313055 + O(7^19)q^{6} + (1254710459798158*7 + O(7^20))*a - 2409975892731410*7 + O(7^20)q^{7} + (13645395825474968 + O(7^20))*a - 26382865636246640 + O(7^20)q^{8} + -(26049323056597189 + O(7^20))*a + 3159716978932714*7 + O(7^20)q^{9} + (1565797003174233 + O(7^19))*a - 72946278900823 + O(7^19)q^{10} + -161051q^{11} + -(28812312361307*7 + O(7^19))*a + 1112425761109889 + O(7^19)q^{12} + -(5290249945585664 + O(7^19))*a + 191659696259903 + O(7^19)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 25739792713100455 + O(7^20)x + 17835578556243587 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^33q^{2} + z^34q^{3} + z^6q^{4} + z^9q^{5} + z^19q^{6} + z^2q^{8} + z^33q^{9} + z^42q^{10} + 5q^{11} + 5q^{12} + z^30q^{13} + \cdots \in S_{12}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^17q^{2} + z^26q^{3} + z^22q^{4} + z^17q^{5} + z^43q^{6} + z^25q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^23)\omega^{4}&0\\0&u(z^25)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(2403633796070289 + O(7^19))*a + 2979174601081107 + O(7^19)q^{3} + (21662035446559841 + O(7^20))*a - 5148414536771278 + O(7^20)q^{4} + -(872951253068868 + O(7^19))*a + 332432666028755*7 + O(7^19)q^{5} + (4906377734381290 + O(7^19))*a - 3806331877795355 + O(7^19)q^{6} + (793451168953439*7 + O(7^20))*a - 208804192747914*7^2 + O(7^20)q^{7} + (12379833008748851 + O(7^20))*a + 2876882332411434 + O(7^20)q^{8} + (11185735187663562 + O(7^20))*a - 504684607040532*7 + O(7^20)q^{9} + -(4586788428634899 + O(7^19))*a + 5129554202989789 + O(7^19)q^{10} + -161051q^{11} + (123098947583271*7 + O(7^19))*a - 5547712083207163 + O(7^19)q^{12} + (5156693005075650 + O(7^19))*a + 4604942271178595 + O(7^19)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -21662035446559841 + O(7^20)x + 5148414536769230 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 4q^{3} + 6q^{4} + 3q^{5} + 3q^{6} + 3q^{8} + 6q^{9} + 4q^{10} + 6q^{11} + 3q^{12} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 3q^{2} + 6q^{3} + 5q^{4} + 2q^{5} + 4q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -11525808211370020 + O(7^20)q^{2} + 2916290527251742 + O(7^19)q^{3} + 1427163846693458 + O(7^20)q^{4} + 1676976543073898 + O(7^19)q^{5} + -9644281299473818 + O(7^20)q^{6} + 626026221346467*7 + O(7^19)q^{7} + 32074643719028767 + O(7^20)q^{8} + 2501352335802795 + O(7^19)q^{9} + -3579614233088400 + O(7^20)q^{10} + -1771561q^{11} + 4901558653528491 + O(7^19)q^{12} + 651732676710568*7^2 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^48q^{2} + z^6q^{3} + z^165q^{4} + z^267q^{5} + z^54q^{6} + z^64q^{8} + z^38q^{9} + z^315q^{10} + q^{11} + 6q^{12} + z^207q^{13} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^276q^{2} + z^291q^{3} + z^279q^{4} + z^324q^{5} + z^225q^{6} + z^190q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^152)\omega^{0}&0\\0&u(z^190)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (23564898988406929 + O(7^20))*a^2 + (23319969183186628 + O(7^20))*a + 446287777056913 + O(7^20)q^{3} + a^2 - 8192q^{4} + (37408620879864044 + O(7^20))*a^2 - (4633362809955459*7 + O(7^20))*a - 790954768982913*7 + O(7^20)q^{5} + (26073395790775999 + O(7^20))*a^2 - (32504652521287725 + O(7^20))*a - 28311397716515978 + O(7^20)q^{6} + (433180026532684*7 + O(7^19))*a^2 + (591007389784562*7 + O(7^19))*a + 33937562169769*7 + O(7^19)q^{7} + -(4905342319407886*7 + O(7^20))*a^2 + (15150576161993628 + O(7^20))*a - 37594680433097990 + O(7^20)q^{8} + (15484323206606431 + O(7^20))*a^2 - (9777854276633943 + O(7^20))*a - 32595562337427093 + O(7^20)q^{9} + (78962326303016*7^3 + O(7^20))*a^2 - (8844806773811312 + O(7^20))*a + 10601817371394682 + O(7^20)q^{10} + 1771561q^{11} + (2876993263679547*7 + O(7^20))*a^2 + (2917317402699678*7 + O(7^20))*a - 39858171996673458 + O(7^20)q^{12} + -(29364245624675578 + O(7^20))*a^2 + (25076818954911898 + O(7^20))*a - 4414131619729502 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 4905342319407886*7 + O(7^20)x^{2} + -15150576162010012 + O(7^20)x + 37594680433097990 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 6q^{3} + 2q^{4} + q^{5} + 2q^{6} + 5q^{9} + 5q^{10} + q^{11} + 5q^{12} + 4q^{13} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 5q^{2} + 6q^{3} + 2q^{4} + q^{5} + 2q^{6} + 5q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(5)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 33517001184949483 + O(7^20)q^{2} + 33231850863240343 + O(7^20)q^{3} + -32058486536294461 + O(7^20)q^{4} + 20294115003009107 + O(7^20)q^{5} + 37470174623551378 + O(7^20)q^{6} + 671106251284312*7 + O(7^19)q^{7} + 3896255426396357*7 + O(7^20)q^{8} + 11661003998567964 + O(7^20)q^{9} + -7262792801023510 + O(7^20)q^{10} + 1771561q^{11} + -2198220083637951 + O(7^20)q^{12} + 3555824576434372 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + z^17q^{2} + z^26q^{3} + z^22q^{4} + z^17q^{5} + z^43q^{6} + z^25q^{7} + z^2q^{8} + z^17q^{9} + z^34q^{10} + 3q^{11} + q^{12} + z^6q^{13} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^23q^{2} + z^38q^{3} + z^10q^{4} + z^23q^{5} + z^13q^{6} + z^31q^{7} + z^14q^{8} + z^23q^{9} + z^46q^{10} + 3q^{11} + q^{12} + z^42q^{13} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 5q^{3} + 2q^{5} + 2q^{6} + q^{8} + 5q^{9} + 5q^{10} + 4q^{11} + \cdots \in S_{10}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + z^186q^{2} + z^237q^{3} + z^333q^{4} + z^144q^{5} + z^81q^{6} + z^76q^{7} + z^58q^{8} + z^38q^{9} + z^330q^{10} + 2q^{11} + 4q^{12} + z^54q^{13} + \cdots \in S_{12}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^222q^{2} + z^327q^{3} + z^243q^{4} + z^216q^{5} + z^207q^{6} + z^304q^{7} + z^106q^{8} + z^152q^{9} + z^96q^{10} + 2q^{11} + 4q^{12} + z^252q^{13} + \cdots \in S_{12}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^276q^{2} + z^291q^{3} + z^279q^{4} + z^324q^{5} + z^225q^{6} + z^190q^{7} + z^64q^{8} + z^266q^{9} + z^258q^{10} + 2q^{11} + 4q^{12} + z^36q^{13} + \cdots \in S_{12}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 6q^{3} + 5q^{4} + 2q^{5} + 4q^{6} + 3q^{7} + 3q^{8} + 3q^{9} + 6q^{10} + 5q^{11} + 2q^{12} + \cdots \in S_{12}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{3} + 5q^{4} + 6q^{5} + 6q^{7} + 6q^{9} + 6q^{11} + 6q^{12} + 3q^{13} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 5q^{4} + 3q^{5} + q^{7} + 5q^{9} + 6q^{11} + 5q^{12} + 3q^{13} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^44q^{2} + z^43q^{3} + 3q^{4} + 5q^{5} + z^39q^{6} + q^{7} + z^44q^{8} + z^31q^{9} + z^36q^{10} + 6q^{11} + z^3q^{12} + z^13q^{13} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 6q^{3} + 2q^{4} + q^{5} + 2q^{6} + 5q^{7} + 5q^{9} + 5q^{10} + q^{11} + 5q^{12} + 4q^{13} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{8} + 6q^{9} + 4q^{10} + q^{11} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 2q^{3} + 6q^{4} + 5q^{5} + 2q^{6} + 6q^{7} + 4q^{8} + q^{9} + 5q^{10} + q^{11} + 5q^{12} + 4q^{13} + \cdots \in S_{14}(\Gamma_0(11);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 12, \quad \ell = 7}$ \quad (5 forms)}

\begin{enumerate}
\item  Consider
$f = q + 5q^{3} + 6q^{5} + 4q^{9} + 4q^{11} + 4q^{13} + \cdots \in S_{12}(\Gamma_0(12);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 3q^{5} + q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 243q^{3} + 2862q^{5} + 1304*7q^{7} + 59049q^{9} + 668196q^{11} + 2052950q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{3} + q^{5} + 4q^{9} + 4q^{11} + 3q^{13} + \cdots \in S_{12}(\Gamma_0(12);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 3q^{5} + q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{4}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -243q^{3} + 9990q^{5} + -12304*7q^{7} + 59049q^{9} + -806004q^{11} + -960250q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{3} + 3q^{5} + q^{7} + 2q^{9} + q^{11} + 4q^{13} + \cdots \in S_{10}(\Gamma_0(12);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{3} + 4q^{5} + 6q^{7} + q^{9} + 2q^{11} + q^{13} + \cdots \in S_{14}(\Gamma_0(12);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + q^{7} + q^{9} + q^{11} + 2q^{13} + \cdots \in S_{14}(\Gamma_0(12);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 13, \quad \ell = 7}$ \quad (33 forms)}

\begin{enumerate}
\item  Consider
$f = q + 6q^{2} + 5q^{3} + 2q^{5} + 2q^{6} + q^{8} + 5q^{9} + 5q^{10} + 6q^{11} + 6q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{2}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 18238764059747678 + O(7^20)q^{2} + -4719551943454812 + O(7^20)q^{3} + 3745269092044937*7 + O(7^20)q^{4} + 14364434234263297 + O(7^20)q^{5} + 756366197430006 + O(7^20)q^{6} + -3904975213256372*7 + O(7^20)q^{7} + -8090606606283109 + O(7^20)q^{8} + -22748373580684873 + O(7^20)q^{9} + 22643564907970256 + O(7^20)q^{10} + -33291813972541027 + O(7^20)q^{11} + 414186205162166*7 + O(7^20)q^{12} + -28561q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^34q^{2} + z^4q^{3} + z^37q^{4} + z^12q^{5} + z^38q^{6} + zq^{8} + 4q^{9} + z^46q^{10} + 2q^{11} + z^41q^{12} + 6q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^2q^{2} + z^36q^{3} + z^21q^{4} + z^28q^{5} + z^38q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(26407275770547997 + O(7^20))*a - 21497043120728541 + O(7^20)q^{3} + (10926801044427675 + O(7^20))*a + 37583490849978439 + O(7^20)q^{4} + -(2409569079113520 + O(7^20))*a - 33628975283469285 + O(7^20)q^{5} + (11644729577740079 + O(7^20))*a - 21494208035613427 + O(7^20)q^{6} + -(701708991362082*7^2 + O(7^20))*a - 925439688618299*7 + O(7^20)q^{7} + (6835984795072813 + O(7^20))*a + 10278889398300043 + O(7^20)q^{8} + (4307782437780713*7 + O(7^20))*a + 38038822815563854 + O(7^20)q^{9} + (14919318776696172 + O(7^20))*a - 17637363433105905 + O(7^20)q^{10} + (3578384245395751*7 + O(7^20))*a - 5147909980767930 + O(7^20)q^{11} + (19513696373731640 + O(7^20))*a + 29422974934658103 + O(7^20)q^{12} + -28561q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -10926801044427675 + O(7^20)x + -37583490849978951 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^46q^{2} + z^28q^{3} + z^19q^{4} + z^36q^{5} + z^26q^{6} + z^7q^{8} + 4q^{9} + z^34q^{10} + 2q^{11} + z^47q^{12} + 6q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^14q^{2} + z^12q^{3} + z^3q^{4} + z^4q^{5} + z^26q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(26407275770547997 + O(7^20))*a - 21497043120728541 + O(7^20)q^{3} + (10926801044427675 + O(7^20))*a + 37583490849978439 + O(7^20)q^{4} + -(2409569079113520 + O(7^20))*a - 33628975283469285 + O(7^20)q^{5} + (11644729577740079 + O(7^20))*a - 21494208035613427 + O(7^20)q^{6} + -(701708991362082*7^2 + O(7^20))*a - 925439688618299*7 + O(7^20)q^{7} + (6835984795072813 + O(7^20))*a + 10278889398300043 + O(7^20)q^{8} + (4307782437780713*7 + O(7^20))*a + 38038822815563854 + O(7^20)q^{9} + (14919318776696172 + O(7^20))*a - 17637363433105905 + O(7^20)q^{10} + (3578384245395751*7 + O(7^20))*a - 5147909980767930 + O(7^20)q^{11} + (19513696373731640 + O(7^20))*a + 29422974934658103 + O(7^20)q^{12} + -28561q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -10926801044427675 + O(7^20)x + -37583490849978951 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^65360q^{2} + z^44032q^{3} + z^68112q^{4} + z^41968q^{5} + z^109392q^{6} + z^110768q^{8} + z^103888q^{9} + z^107328q^{10} + z^67424q^{11} + z^112144q^{12} + q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^304q^{2} + z^14q^{3} + z^84q^{4} + z^236q^{5} + z^318q^{6} + z^71q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^271)\omega^{1}&0\\0&u(z^71)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(1077691251316388 + O(7^20))*a^2 + (31241474456528662 + O(7^20))*a + 38749208152044498 + O(7^20)q^{3} + a^2 - 512q^{4} + -(12135981513572536 + O(7^20))*a^2 + (9620833608511015 + O(7^20))*a - 3771160044357978*7 + O(7^20)q^{5} + (8392678051344892 + O(7^20))*a^2 - (1485726033938591 + O(7^20))*a - 11253048580188871 + O(7^20)q^{6} + (1450989311459988*7 + O(7^20))*a^2 + (2836750518682972*7 + O(7^20))*a + 1611869847230560*7 + O(7^20)q^{7} + -(374868796274107*7^2 + O(7^20))*a^2 - (38355287372638980 + O(7^20))*a + 6313684964894703 + O(7^20)q^{8} + (22040599890190837 + O(7^20))*a^2 + (31303957070841656 + O(7^20))*a + 30865658895888175 + O(7^20)q^{9} + (26813774858682205 + O(7^20))*a^2 - (55356017523619*7^3 + O(7^20))*a + 15682896395583688 + O(7^20)q^{10} + -(696043347116974 + O(7^20))*a^2 + (3854098865074884 + O(7^20))*a + 30324208109128348 + O(7^20)q^{11} + -(36542067066020378 + O(7^20))*a^2 + (24849727955094926 + O(7^20))*a - 32971616367268097 + O(7^20)q^{12} + 28561q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 374868796274107*7^2 + O(7^20)x^{2} + 5479326767519708*7 + O(7^20)x + -6313684964894703 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^104576q^{2} + z^72928q^{3} + z^6192q^{4} + z^58480q^{5} + z^59856q^{6} + z^69488q^{8} + z^21328q^{9} + z^45408q^{10} + z^1376q^{11} + z^79120q^{12} + q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^76q^{2} + z^98q^{3} + z^246q^{4} + z^284q^{5} + z^174q^{6} + z^155q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^187)\omega^{1}&0\\0&u(z^155)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(1077691251316388 + O(7^20))*a^2 + (31241474456528662 + O(7^20))*a + 38749208152044498 + O(7^20)q^{3} + a^2 - 512q^{4} + -(12135981513572536 + O(7^20))*a^2 + (9620833608511015 + O(7^20))*a - 3771160044357978*7 + O(7^20)q^{5} + (8392678051344892 + O(7^20))*a^2 - (1485726033938591 + O(7^20))*a - 11253048580188871 + O(7^20)q^{6} + (1450989311459988*7 + O(7^20))*a^2 + (2836750518682972*7 + O(7^20))*a + 1611869847230560*7 + O(7^20)q^{7} + -(374868796274107*7^2 + O(7^20))*a^2 - (38355287372638980 + O(7^20))*a + 6313684964894703 + O(7^20)q^{8} + (22040599890190837 + O(7^20))*a^2 + (31303957070841656 + O(7^20))*a + 30865658895888175 + O(7^20)q^{9} + (26813774858682205 + O(7^20))*a^2 - (55356017523619*7^3 + O(7^20))*a + 15682896395583688 + O(7^20)q^{10} + -(696043347116974 + O(7^20))*a^2 + (3854098865074884 + O(7^20))*a + 30324208109128348 + O(7^20)q^{11} + -(36542067066020378 + O(7^20))*a^2 + (24849727955094926 + O(7^20))*a - 32971616367268097 + O(7^20)q^{12} + 28561q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 374868796274107*7^2 + O(7^20)x^{2} + 5479326767519708*7 + O(7^20)x + -6313684964894703 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^26144q^{2} + z^39904q^{3} + z^43344q^{4} + z^56416q^{5} + z^66048q^{6} + z^15824q^{8} + z^31648q^{9} + z^82560q^{10} + z^9632q^{11} + z^83248q^{12} + q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^190q^{2} + z^2q^{3} + z^12q^{4} + z^278q^{5} + z^192q^{6} + z^59q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^283)\omega^{1}&0\\0&u(z^59)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(1077691251316388 + O(7^20))*a^2 + (31241474456528662 + O(7^20))*a + 38749208152044498 + O(7^20)q^{3} + a^2 - 512q^{4} + -(12135981513572536 + O(7^20))*a^2 + (9620833608511015 + O(7^20))*a - 3771160044357978*7 + O(7^20)q^{5} + (8392678051344892 + O(7^20))*a^2 - (1485726033938591 + O(7^20))*a - 11253048580188871 + O(7^20)q^{6} + (1450989311459988*7 + O(7^20))*a^2 + (2836750518682972*7 + O(7^20))*a + 1611869847230560*7 + O(7^20)q^{7} + -(374868796274107*7^2 + O(7^20))*a^2 - (38355287372638980 + O(7^20))*a + 6313684964894703 + O(7^20)q^{8} + (22040599890190837 + O(7^20))*a^2 + (31303957070841656 + O(7^20))*a + 30865658895888175 + O(7^20)q^{9} + (26813774858682205 + O(7^20))*a^2 - (55356017523619*7^3 + O(7^20))*a + 15682896395583688 + O(7^20)q^{10} + -(696043347116974 + O(7^20))*a^2 + (3854098865074884 + O(7^20))*a + 30324208109128348 + O(7^20)q^{11} + -(36542067066020378 + O(7^20))*a^2 + (24849727955094926 + O(7^20))*a - 32971616367268097 + O(7^20)q^{12} + 28561q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 374868796274107*7^2 + O(7^20)x^{2} + 5479326767519708*7 + O(7^20)x + -6313684964894703 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 5q^{4} + 4q^{8} + 2q^{9} + q^{11} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 3q^{4} + q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -36355254830082583 + O(7^20)q^{2} + 3196627519955022*7 + O(7^20)q^{3} + 29346106439894291 + O(7^20)q^{4} + 298588357548281*7 + O(7^20)q^{5} + 1708032065327724*7 + O(7^20)q^{6} + -4284339829459093*7 + O(7^20)q^{7} + -33300029093768685 + O(7^20)q^{8} + -1035727297615050 + O(7^20)q^{9} + -2705438795536675*7 + O(7^20)q^{10} + -28621322355903654 + O(7^20)q^{11} + 2023042756413492*7 + O(7^20)q^{12} + 371293q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^17q^{2} + z^45q^{3} + z^20q^{4} + z^3q^{5} + z^14q^{6} + z^39q^{8} + z^28q^{9} + z^20q^{10} + z^26q^{11} + z^17q^{12} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + zq^{2} + z^37q^{3} + z^36q^{4} + z^11q^{5} + z^38q^{6} + z^37q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^11)\omega^{4}&0\\0&u(z^37)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(28021382744190732 + O(7^20))*a + 29075221695248335 + O(7^20)q^{3} + (37062850254546356 + O(7^20))*a - 25336027171839197 + O(7^20)q^{4} + (9137506709279255 + O(7^20))*a + 21130329446418833 + O(7^20)q^{5} + (27050151177395587 + O(7^20))*a - 8588893950849347 + O(7^20)q^{6} + -(679107457631557*7 + O(7^20))*a - 1532777315004389*7 + O(7^20)q^{7} + -(3196942254541653 + O(7^20))*a + 3685562706407208 + O(7^20)q^{8} + -(20440276081116032 + O(7^20))*a + 3827795802700185 + O(7^20)q^{9} + -(11678943798541983 + O(7^20))*a + 35581260741374843 + O(7^20)q^{10} + (21274710360332768 + O(7^20))*a - 31856268635061651 + O(7^20)q^{11} + (30888815272483575 + O(7^20))*a - 1454140763125364*7 + O(7^20)q^{12} + 371293q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -37062850254546356 + O(7^20)x + 25336027171837149 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^23q^{2} + z^27q^{3} + z^44q^{4} + z^21q^{5} + z^2q^{6} + z^33q^{8} + z^4q^{9} + z^44q^{10} + z^38q^{11} + z^23q^{12} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + z^19q^{3} + z^12q^{4} + z^29q^{5} + z^26q^{6} + z^19q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{4}&0\\0&u(z^19)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(28021382744190732 + O(7^20))*a + 29075221695248335 + O(7^20)q^{3} + (37062850254546356 + O(7^20))*a - 25336027171839197 + O(7^20)q^{4} + (9137506709279255 + O(7^20))*a + 21130329446418833 + O(7^20)q^{5} + (27050151177395587 + O(7^20))*a - 8588893950849347 + O(7^20)q^{6} + -(679107457631557*7 + O(7^20))*a - 1532777315004389*7 + O(7^20)q^{7} + -(3196942254541653 + O(7^20))*a + 3685562706407208 + O(7^20)q^{8} + -(20440276081116032 + O(7^20))*a + 3827795802700185 + O(7^20)q^{9} + -(11678943798541983 + O(7^20))*a + 35581260741374843 + O(7^20)q^{10} + (21274710360332768 + O(7^20))*a - 31856268635061651 + O(7^20)q^{11} + (30888815272483575 + O(7^20))*a - 1454140763125364*7 + O(7^20)q^{12} + 371293q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -37062850254546356 + O(7^20)x + 25336027171837149 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 5q^{4} + 4q^{8} + 2q^{9} + q^{11} + q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 3q^{4} + q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -16445726030701936 + O(7^20)q^{2} + 4189107182921369*7 + O(7^20)q^{3} + -36544267013703472 + O(7^20)q^{4} + -1673173333360570*7 + O(7^20)q^{5} + 3818825363746450*7 + O(7^20)q^{6} + -3763255715045811*7 + O(7^20)q^{7} + -39519223426377040 + O(7^20)q^{8} + -17828849065823878 + O(7^20)q^{9} + -1231661461880160*7 + O(7^20)q^{10} + 34753324923820259 + O(7^20)q^{11} + -492489757187607*7 + O(7^20)q^{12} + -371293q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^41667q^{2} + z^51471q^{3} + z^49020q^{4} + z^66177q^{5} + z^93138q^{6} + z^95589q^{8} + z^68628q^{9} + z^107844q^{10} + z^63726q^{11} + z^100491q^{12} + q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + zq^{2} + z^37q^{3} + z^36q^{4} + z^11q^{5} + z^38q^{6} + z^37q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^11)\omega^{1}&0\\0&u(z^37)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (24858059536616482 + O(7^20))*a + 27577079197165171 + O(7^20)q^{3} + (37448322846586752 + O(7^20))*a - 3235560941186296 + O(7^20)q^{4} + -(26981113215177841 + O(7^20))*a - 27410988609564893 + O(7^20)q^{5} + -(21933428747839745 + O(7^20))*a - 36218700187818853 + O(7^20)q^{6} + (1487600004904868*7 + O(7^20))*a + 5181527171766606*7 + O(7^20)q^{7} + (9098703365898928 + O(7^20))*a - 14135833915540886 + O(7^20)q^{8} + (33372596272073750 + O(7^20))*a - 32780654801961352 + O(7^20)q^{9} + (2912222458089271 + O(7^20))*a + 14381002453515553 + O(7^20)q^{10} + -(39721507954084261 + O(7^20))*a - 6004586478496232 + O(7^20)q^{11} + -(21394170555465912 + O(7^20))*a + 902583619638053*7 + O(7^20)q^{12} + -371293q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -37448322846586752 + O(7^20)x + 3235560941184248 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^56373q^{2} + z^7353q^{3} + z^107844q^{4} + z^110295q^{5} + z^63726q^{6} + z^80883q^{8} + z^9804q^{9} + z^49020q^{10} + z^93138q^{11} + z^115197q^{12} + q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^7q^{2} + z^19q^{3} + z^12q^{4} + z^29q^{5} + z^26q^{6} + z^19q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{1}&0\\0&u(z^19)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (24858059536616482 + O(7^20))*a + 27577079197165171 + O(7^20)q^{3} + (37448322846586752 + O(7^20))*a - 3235560941186296 + O(7^20)q^{4} + -(26981113215177841 + O(7^20))*a - 27410988609564893 + O(7^20)q^{5} + -(21933428747839745 + O(7^20))*a - 36218700187818853 + O(7^20)q^{6} + (1487600004904868*7 + O(7^20))*a + 5181527171766606*7 + O(7^20)q^{7} + (9098703365898928 + O(7^20))*a - 14135833915540886 + O(7^20)q^{8} + (33372596272073750 + O(7^20))*a - 32780654801961352 + O(7^20)q^{9} + (2912222458089271 + O(7^20))*a + 14381002453515553 + O(7^20)q^{10} + -(39721507954084261 + O(7^20))*a - 6004586478496232 + O(7^20)q^{11} + -(21394170555465912 + O(7^20))*a + 902583619638053*7 + O(7^20)q^{12} + -371293q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -37448322846586752 + O(7^20)x + 3235560941184248 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^190q^{2} + z^155q^{3} + z^132q^{4} + z^227q^{5} + z^3q^{6} + z^202q^{8} + z^290q^{9} + z^75q^{10} + z^118q^{11} + z^287q^{12} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^76q^{2} + z^98q^{3} + z^246q^{4} + z^284q^{5} + z^174q^{6} + z^155q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^187)\omega^{0}&0\\0&u(z^155)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (23564704238604447 + O(7^20))*a^2 - (37477698527152809 + O(7^20))*a + 10063242979415074 + O(7^20)q^{3} + a^2 - 8192q^{4} + (20136000030785542 + O(7^20))*a^2 - (19990868132239370 + O(7^20))*a + 3454282104617609*7 + O(7^20)q^{5} + (20777838790460260 + O(7^20))*a^2 + (27504583131820866 + O(7^20))*a - 31443928158816109 + O(7^20)q^{6} + -(5555028395258113*7 + O(7^20))*a^2 + (891278119877658*7 + O(7^20))*a - 2890923861855912*7 + O(7^20)q^{7} + (506870939569256*7^2 + O(7^20))*a^2 + (11773863848622179 + O(7^20))*a - 38691889953301651 + O(7^20)q^{8} + -(33263274134552832 + O(7^20))*a^2 + (27430308738465673 + O(7^20))*a + 11920500666346210 + O(7^20)q^{9} + (32202090275955164 + O(7^20))*a^2 - (375819806480880*7 + O(7^20))*a - 7329482726004120 + O(7^20)q^{10} + -(21396170670304244 + O(7^20))*a^2 - (16517401540426420 + O(7^20))*a + 17840341826496872 + O(7^20)q^{11} + -(22102840798298102 + O(7^20))*a^2 + (34772712092014713 + O(7^20))*a - 16618992163264095 + O(7^20)q^{12} + -4826809q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -506870939569256*7^2 + O(7^20)x^{2} + -1681980549805509*7 + O(7^20)x + 38691889953301651 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^304q^{2} + z^59q^{3} + z^240q^{4} + z^221q^{5} + z^21q^{6} + z^46q^{8} + z^320q^{9} + z^183q^{10} + z^142q^{11} + z^299q^{12} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^190q^{2} + z^2q^{3} + z^12q^{4} + z^278q^{5} + z^192q^{6} + z^59q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^283)\omega^{0}&0\\0&u(z^59)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (23564704238604447 + O(7^20))*a^2 - (37477698527152809 + O(7^20))*a + 10063242979415074 + O(7^20)q^{3} + a^2 - 8192q^{4} + (20136000030785542 + O(7^20))*a^2 - (19990868132239370 + O(7^20))*a + 3454282104617609*7 + O(7^20)q^{5} + (20777838790460260 + O(7^20))*a^2 + (27504583131820866 + O(7^20))*a - 31443928158816109 + O(7^20)q^{6} + -(5555028395258113*7 + O(7^20))*a^2 + (891278119877658*7 + O(7^20))*a - 2890923861855912*7 + O(7^20)q^{7} + (506870939569256*7^2 + O(7^20))*a^2 + (11773863848622179 + O(7^20))*a - 38691889953301651 + O(7^20)q^{8} + -(33263274134552832 + O(7^20))*a^2 + (27430308738465673 + O(7^20))*a + 11920500666346210 + O(7^20)q^{9} + (32202090275955164 + O(7^20))*a^2 - (375819806480880*7 + O(7^20))*a - 7329482726004120 + O(7^20)q^{10} + -(21396170670304244 + O(7^20))*a^2 - (16517401540426420 + O(7^20))*a + 17840341826496872 + O(7^20)q^{11} + -(22102840798298102 + O(7^20))*a^2 + (34772712092014713 + O(7^20))*a - 16618992163264095 + O(7^20)q^{12} + -4826809q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -506870939569256*7^2 + O(7^20)x^{2} + -1681980549805509*7 + O(7^20)x + 38691889953301651 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^76q^{2} + z^71q^{3} + z^312q^{4} + z^179q^{5} + z^147q^{6} + z^322q^{8} + z^188q^{9} + z^255q^{10} + z^310q^{11} + z^41q^{12} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^304q^{2} + z^14q^{3} + z^84q^{4} + z^236q^{5} + z^318q^{6} + z^71q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^271)\omega^{0}&0\\0&u(z^71)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (23564704238604447 + O(7^20))*a^2 - (37477698527152809 + O(7^20))*a + 10063242979415074 + O(7^20)q^{3} + a^2 - 8192q^{4} + (20136000030785542 + O(7^20))*a^2 - (19990868132239370 + O(7^20))*a + 3454282104617609*7 + O(7^20)q^{5} + (20777838790460260 + O(7^20))*a^2 + (27504583131820866 + O(7^20))*a - 31443928158816109 + O(7^20)q^{6} + -(5555028395258113*7 + O(7^20))*a^2 + (891278119877658*7 + O(7^20))*a - 2890923861855912*7 + O(7^20)q^{7} + (506870939569256*7^2 + O(7^20))*a^2 + (11773863848622179 + O(7^20))*a - 38691889953301651 + O(7^20)q^{8} + -(33263274134552832 + O(7^20))*a^2 + (27430308738465673 + O(7^20))*a + 11920500666346210 + O(7^20)q^{9} + (32202090275955164 + O(7^20))*a^2 - (375819806480880*7 + O(7^20))*a - 7329482726004120 + O(7^20)q^{10} + -(21396170670304244 + O(7^20))*a^2 - (16517401540426420 + O(7^20))*a + 17840341826496872 + O(7^20)q^{11} + -(22102840798298102 + O(7^20))*a^2 + (34772712092014713 + O(7^20))*a - 16618992163264095 + O(7^20)q^{12} + -4826809q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -506870939569256*7^2 + O(7^20)x^{2} + -1681980549805509*7 + O(7^20)x + 38691889953301651 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^73530q^{2} + z^49020q^{3} + z^85785q^{4} + z^107844q^{5} + z^4902q^{6} + z^17157q^{8} + 2q^{9} + z^63726q^{10} + 4q^{11} + z^17157q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^14q^{2} + z^12q^{3} + z^3q^{4} + z^4q^{5} + z^26q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (7717665980667858 + O(7^20))*a - 10691423681325054 + O(7^20)q^{3} + -(5879378750302886 + O(7^20))*a - 28878309303152417 + O(7^20)q^{4} + -(9136772978480194 + O(7^20))*a + 33793478753135503 + O(7^20)q^{5} + -(32752493017622662 + O(7^20))*a + 21561179967463383 + O(7^20)q^{6} + -(763521939201171*7^2 + O(7^20))*a + 4651957060170795*7 + O(7^20)q^{7} + (33099161618614809 + O(7^20))*a + 12146217013474267 + O(7^20)q^{8} + -(542242030239460*7^2 + O(7^20))*a + 39470250908885288 + O(7^20)q^{9} + -(35481560194173971 + O(7^20))*a - 17763112971085554 + O(7^20)q^{10} + -(105368760990832*7^3 + O(7^20))*a + 31695680287411368 + O(7^20)q^{11} + -(5297203016040876 + O(7^20))*a - 28978451731769733 + O(7^20)q^{12} + 4826809q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 5879378750302886 + O(7^20)x + 28878309303144225 + O(7^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{8} + 6q^{9} + 4q^{10} + 5q^{11} + q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 10743948263694852 + O(7^20)q^{2} + -20063411012620230 + O(7^20)q^{3} + -4333152687900405*7 + O(7^20)q^{4} + 27724218787470519 + O(7^20)q^{5} + 4130640570972466 + O(7^20)q^{6} + -4459075454800968*7 + O(7^20)q^{7} + -25704989492975778 + O(7^20)q^{8} + -8413093252907344 + O(7^20)q^{9} + 28850283146458778 + O(7^20)q^{10} + -17125835957858259 + O(7^20)q^{11} + -1847911340498411*7 + O(7^20)q^{12} + 4826809q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^44118q^{2} + z^107844q^{3} + z^12255q^{4} + z^49020q^{5} + z^34314q^{6} + z^2451q^{8} + 2q^{9} + z^93138q^{10} + 4q^{11} + z^2451q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^2q^{2} + z^36q^{3} + z^21q^{4} + z^28q^{5} + z^38q^{6} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (7717665980667858 + O(7^20))*a - 10691423681325054 + O(7^20)q^{3} + -(5879378750302886 + O(7^20))*a - 28878309303152417 + O(7^20)q^{4} + -(9136772978480194 + O(7^20))*a + 33793478753135503 + O(7^20)q^{5} + -(32752493017622662 + O(7^20))*a + 21561179967463383 + O(7^20)q^{6} + -(763521939201171*7^2 + O(7^20))*a + 4651957060170795*7 + O(7^20)q^{7} + (33099161618614809 + O(7^20))*a + 12146217013474267 + O(7^20)q^{8} + -(542242030239460*7^2 + O(7^20))*a + 39470250908885288 + O(7^20)q^{9} + -(35481560194173971 + O(7^20))*a - 17763112971085554 + O(7^20)q^{10} + -(105368760990832*7^3 + O(7^20))*a + 31695680287411368 + O(7^20)q^{11} + -(5297203016040876 + O(7^20))*a - 28978451731769733 + O(7^20)q^{12} + 4826809q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 5879378750302886 + O(7^20)x + 28878309303144225 + O(7^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 3q^{4} + q^{7} + 4q^{8} + q^{9} + 2q^{11} + 6q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2451q^{2} + z^90687q^{3} + z^88236q^{4} + z^26961q^{5} + z^93138q^{6} + z^90687q^{7} + z^95589q^{8} + z^29412q^{9} + z^29412q^{10} + z^102942q^{11} + z^61275q^{12} + q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^17157q^{2} + z^46569q^{3} + z^29412q^{4} + z^71079q^{5} + z^63726q^{6} + z^46569q^{7} + z^80883q^{8} + z^88236q^{9} + z^88236q^{10} + z^14706q^{11} + z^75981q^{12} + q^{13} + \cdots \in S_{10}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^2q^{2} + z^36q^{3} + z^21q^{4} + z^28q^{5} + z^38q^{6} + 3q^{7} + zq^{8} + q^{9} + z^30q^{10} + q^{11} + z^9q^{12} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^14q^{2} + z^12q^{3} + z^3q^{4} + z^4q^{5} + z^26q^{6} + 3q^{7} + z^7q^{8} + q^{9} + z^18q^{10} + q^{11} + z^15q^{12} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^65360q^{2} + z^688q^{3} + z^4128q^{4} + z^95632q^{5} + z^66048q^{6} + z^20296q^{7} + z^15824q^{8} + z^70864q^{9} + z^43344q^{10} + z^88064q^{11} + z^4816q^{12} + q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^104576q^{2} + z^4816q^{3} + z^28896q^{4} + z^81184q^{5} + z^109392q^{6} + z^24424q^{7} + z^110768q^{8} + z^25456q^{9} + z^68112q^{10} + z^28208q^{11} + z^33712q^{12} + q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^26144q^{2} + z^33712q^{3} + z^84624q^{4} + z^97696q^{5} + z^59856q^{6} + z^53320q^{7} + z^69488q^{8} + z^60544q^{9} + z^6192q^{10} + z^79808q^{11} + z^688q^{12} + q^{13} + \cdots \in S_{12}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 3q^{3} + 5q^{6} + q^{7} + 6q^{8} + 6q^{9} + 2q^{11} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 2q^{4} + 4q^{5} + 6q^{7} + 4q^{9} + 6q^{10} + q^{11} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{7} + q^{8} + 6q^{9} + 4q^{10} + 5q^{11} + 6q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{3} + 6x^{2} + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{3} + 5q^{4} + 4q^{5} + q^{7} + q^{9} + 4q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^80152q^{2} + z^72240q^{3} + z^65016q^{4} + z^113176q^{5} + z^34744q^{6} + 6q^{7} + z^65016q^{8} + z^38872q^{9} + z^75680q^{10} + z^13416q^{11} + 3q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^90472q^{2} + z^35088q^{3} + z^102168q^{4} + z^86344q^{5} + z^7912q^{6} + 6q^{7} + z^102168q^{8} + z^36808q^{9} + z^59168q^{10} + z^93912q^{11} + 3q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^45064q^{2} + z^10320q^{3} + z^9288q^{4} + z^16168q^{5} + z^55384q^{6} + 6q^{7} + z^9288q^{8} + z^22360q^{9} + z^61232q^{10} + z^69144q^{11} + 3q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 15, \quad \ell = 7}$ \quad (21 forms)}

\begin{enumerate}
\item  Consider
$f = q + 4q^{3} + 6q^{4} + 5q^{5} + 2q^{9} + 3q^{12} + 6q^{13} + \cdots \in S_{10}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{3} + 3q^{4} + 3q^{5} + 5q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(5)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4724792588377961*7 + O(7^20)q^{2} + 81q^{3} + -9940716126626615 + O(7^20)q^{4} + -625q^{5} + -4854236644072021*7 + O(7^20)q^{6} + -25211586056330585*7 + O(7^21)q^{7} + -2067322582271221*7 + O(7^20)q^{8} + 6561q^{9} + -681514724581588*7 + O(7^20)q^{10} + -1088927546902001*7 + O(7^20)q^{11} + -7275343280635805 + O(7^20)q^{12} + 26970258623687392 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 4q^{3} + 3q^{4} + 5q^{5} + 6q^{6} + 3q^{8} + 2q^{9} + 4q^{10} + 5q^{11} + 5q^{12} + 2q^{13} + \cdots \in S_{10}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 6q^{2} + 6q^{3} + 6q^{4} + q^{5} + q^{6} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{9}$. The form
$f_0 = q + -33073548118645708 + O(7^20)q^{2} + 81q^{3} + 9940716126628136 + O(7^20)q^{4} + -625q^{5} + 33979656508505686 + O(7^20)q^{6} + 3601655150904127*7^2 + O(7^21)q^{7} + 14471258075948194 + O(7^20)q^{8} + 6561q^{9} + 4770603072059241 + O(7^20)q^{10} + 7622492828349495 + O(7^20)q^{11} + 7275343280759006 + O(7^20)q^{12} + -26970258623543716 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + z^9q^{2} + 3q^{3} + z^17q^{4} + 2q^{5} + z^17q^{6} + z^13q^{8} + 2q^{9} + z^25q^{10} + z^31q^{11} + z^25q^{12} + z^45q^{13} + \cdots \in S_{10}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^25q^{2} + 5q^{3} + zq^{4} + 4q^{5} + z^17q^{6} + z^4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^44)\omega^{1}&0\\0&u(z^4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -81q^{3} + 31*a - 30*7q^{4} + 625q^{5} + -81*aq^{6} + 32*7*a + 512*7q^{7} + 239*a + 9362q^{8} + 6561q^{9} + 625*aq^{10} + -2368*a + 25948q^{11} + -2511*a + 2430*7q^{12} + -5344*a + 94974q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -31x + -302$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^15q^{2} + 3q^{3} + z^23q^{4} + 2q^{5} + z^23q^{6} + z^43q^{8} + 2q^{9} + z^31q^{10} + z^25q^{11} + z^31q^{12} + z^27q^{13} + \cdots \in S_{10}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^31q^{2} + 5q^{3} + z^7q^{4} + 4q^{5} + z^23q^{6} + z^28q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^20)\omega^{1}&0\\0&u(z^28)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -81q^{3} + 31*a - 30*7q^{4} + 625q^{5} + -81*aq^{6} + 32*7*a + 512*7q^{7} + 239*a + 9362q^{8} + 6561q^{9} + 625*aq^{10} + -2368*a + 25948q^{11} + -2511*a + 2430*7q^{12} + -5344*a + 94974q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -31x + -302$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 2q^{3} + 4q^{4} + 4q^{5} + 5q^{6} + 4q^{9} + 3q^{10} + 2q^{11} + q^{12} + 4q^{13} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 4q^{3} + q^{4} + 2q^{5} + 5q^{6} + 6q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 6986710225391042 + O(7^20)q^{2} + -243q^{3} + -11034966632473185 + O(7^20)q^{4} + -3125q^{5} + -22132992520171185 + O(7^20)q^{6} + -13456347992322657*7 + O(7^21)q^{7} + -1459323355226542*7 + O(7^20)q^{8} + 59049q^{9} + 29611511198682024 + O(7^20)q^{10} + 23709742589519408 + O(7^20)q^{11} + -31440162427824079 + O(7^20)q^{12} + 15498634927858149 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + 4q^{5} + 4q^{6} + 6q^{8} + 4q^{9} + q^{10} + 5q^{11} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{2} + 3q^{3} + 5q^{5} + 3q^{6} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{4}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -6986710225391055 + O(7^20)q^{2} + -243q^{3} + 1576423804638582*7 + O(7^20)q^{4} + -3125q^{5} + 22132992520174344 + O(7^20)q^{6} + 13456347992323769*7 + O(7^21)q^{7} + 10215263486620933 + O(7^20)q^{8} + 59049q^{9} + -29611511198641399 + O(7^20)q^{10} + -23709742589223840 + O(7^20)q^{11} + 4491451775511436*7 + O(7^20)q^{12} + -15498634927200657 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 5q^{3} + 4q^{4} + 3q^{5} + 2q^{6} + 4q^{9} + 4q^{10} + 2q^{11} + 6q^{12} + 3q^{13} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 4q^{3} + q^{4} + 2q^{5} + 5q^{6} + 6q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{4}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 17016513340204580 + O(7^20)q^{2} + 243q^{3} + -29082468024135704 + O(7^20)q^{4} + 3125q^{5} + -14185105806111112 + O(7^20)q^{6} + 5396370061046494*7 + O(7^20)q^{7} + 193399401046111*7^2 + O(7^20)q^{8} + 59049q^{9} + 34954833929719834 + O(7^20)q^{10} + 34266114840756507 + O(7^20)q^{11} + 34471970622492017 + O(7^20)q^{12} + 17887534118948637 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 5q^{3} + 3q^{5} + 3q^{6} + 6q^{8} + 4q^{9} + 6q^{10} + 5q^{11} + q^{13} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{2} + 3q^{3} + 5q^{5} + 3q^{6} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(4)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 7331023606406831 + O(7^20)q^{2} + 243q^{3} + 1536937681856897*7 + O(7^20)q^{4} + 3125q^{5} + 26008877809395911 + O(7^20)q^{6} + -4044974932987831*7 + O(7^20)q^{7} + 21206378397953475 + O(7^20)q^{8} + 59049q^{9} + 9068342606702588 + O(7^20)q^{10} + 11333354898510447 + O(7^20)q^{11} + -2687684426087748*7 + O(7^20)q^{12} + -14865760551194923 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^41q^{2} + q^{3} + z^33q^{4} + 6q^{5} + z^41q^{6} + z^13q^{8} + q^{9} + z^17q^{10} + z^47q^{11} + z^33q^{12} + z^21q^{13} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^25q^{2} + 5q^{3} + zq^{4} + 4q^{5} + z^17q^{6} + z^4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^44)\omega^{0}&0\\0&u(z^4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 729q^{3} + 131*a - 780*7q^{4} + -15625q^{5} + 729*aq^{6} + 232*7*a + 20252*7q^{7} + 3509*a + 357892q^{8} + 531441q^{9} + -15625*aq^{10} + 35552*a + 794288q^{11} + 95499*a - 568620*7q^{12} + 3016*a + 683034q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -131x + -2732$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^47q^{2} + q^{3} + z^39q^{4} + 6q^{5} + z^47q^{6} + z^43q^{8} + q^{9} + z^23q^{10} + z^41q^{11} + z^39q^{12} + z^3q^{13} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^31q^{2} + 5q^{3} + z^7q^{4} + 4q^{5} + z^23q^{6} + z^28q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^20)\omega^{0}&0\\0&u(z^28)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 729q^{3} + 131*a - 780*7q^{4} + -15625q^{5} + 729*aq^{6} + 232*7*a + 20252*7q^{7} + 3509*a + 357892q^{8} + 531441q^{9} + -15625*aq^{10} + 35552*a + 794288q^{11} + 95499*a - 568620*7q^{12} + 3016*a + 683034q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -131x + -2732$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{3} + 5q^{4} + q^{5} + q^{9} + 2q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{3} + 3q^{4} + 3q^{5} + 5q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{0}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 5189973582128632*7 + O(7^20)q^{2} + -729q^{3} + 45309890105238015 + O(7^21)q^{4} + 15625q^{5} + 942460172110748*7 + O(7^20)q^{6} + -24826402549611*7 + O(7^19)q^{7} + 1872658279172179*7 + O(7^20)q^{8} + 531441q^{9} + 1596872015335698*7 + O(7^20)q^{10} + -335192499846327*7 + O(7^20)q^{11} + -76703905804756522 + O(7^21)q^{12} + -31450250938702742 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 6q^{3} + 6q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 6q^{10} + 3q^{11} + q^{12} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 6q^{2} + 6q^{3} + 6q^{4} + q^{5} + q^{6} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + aq^{2} + -729q^{3} + -(36329815074900489 + O(7^20))*a - 12979881045403891 + O(7^20)q^{4} + 15625q^{5} + -729*aq^{6} + -(21798591536378883*7^-1 + O(7^19))*a - 3273065291826409*7^-1 + O(7^19)q^{7} + -(32330009059826591 + O(7^20))*a - 34022661322911448 + O(7^20)q^{8} + 531441q^{9} + 15625*aq^{10} + (21332298747079424 + O(7^20))*a + 20758949022325523 + O(7^20)q^{11} + -(6597221204727851 + O(7^20))*a - 32946407316391580 + O(7^20)q^{12} + (20924590739614594 + O(7^20))*a - 15244769172240830 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 36329815074900489 + O(7^20)x + 12979881045395699 + O(7^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{2} + 4q^{3} + q^{4} + 2q^{5} + 5q^{6} + 6q^{7} + 2q^{9} + 6q^{10} + 4q^{11} + 4q^{12} + 4q^{13} + \cdots \in S_{10}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 3q^{3} + 5q^{5} + 3q^{6} + 4q^{7} + 6q^{8} + 2q^{9} + 5q^{10} + 3q^{11} + q^{13} + \cdots \in S_{10}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + 3q^{4} + 3q^{5} + 5q^{7} + 4q^{9} + 6q^{12} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^25q^{2} + 5q^{3} + zq^{4} + 4q^{5} + z^17q^{6} + z^4q^{7} + z^13q^{8} + 4q^{9} + z^9q^{10} + z^15q^{11} + z^41q^{12} + z^45q^{13} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^31q^{2} + 5q^{3} + z^7q^{4} + 4q^{5} + z^23q^{6} + z^28q^{7} + z^43q^{8} + 4q^{9} + z^15q^{10} + z^9q^{11} + z^47q^{12} + z^27q^{13} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 5q^{3} + 3q^{5} + 4q^{6} + q^{7} + q^{8} + 4q^{9} + q^{10} + 3q^{11} + \cdots \in S_{12}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + 6q^{4} + q^{5} + q^{6} + q^{7} + 4q^{8} + q^{9} + q^{10} + 6q^{12} + q^{13} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^44q^{2} + 6q^{3} + 3q^{4} + 6q^{5} + z^20q^{6} + q^{7} + z^44q^{8} + q^{9} + z^20q^{10} + z^11q^{11} + 4q^{12} + z^36q^{13} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^20q^{2} + 6q^{3} + 3q^{4} + 6q^{5} + z^44q^{6} + q^{7} + z^20q^{8} + q^{9} + z^44q^{10} + z^29q^{11} + 4q^{12} + z^12q^{13} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{7}).$
Also,~$z$ satisfies the equation $x^{2} + 6x + 3=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 16, \quad \ell = 7}$ \quad (15 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{3} + 5q^{5} + 5q^{9} + 4q^{11} + 2q^{13} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{3} + 3q^{5} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -68q^{3} + 1510q^{5} + -1464*7q^{7} + -15059q^{9} + -3916q^{11} + -176594q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{3} + 6q^{5} + 3q^{9} + 5q^{11} + 2q^{13} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 5q^{3} + 5q^{5} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -228q^{3} + -666q^{5} + 904*7q^{7} + 32301q^{9} + 30420q^{11} + -32338q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{3} + 4q^{5} + 6q^{9} + q^{11} + 6q^{13} + \cdots \in S_{12}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 5q^{5} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{4}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -26125321683010552 + O(7^20)q^{3} + 5665204994325650 + O(7^20)q^{5} + 2832602497154354*7 + O(7^21)q^{7} + -212619578000992576 + O(7^21)q^{9} + 231380275661125368 + O(7^21)q^{11} + 31202403234781680 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{3} + 3q^{5} + 6q^{9} + q^{11} + q^{13} + \cdots \in S_{12}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 5q^{5} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(4)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 26125321683010496 + O(7^20)q^{3} + -5665204994317782 + O(7^20)q^{5} + -2832602497167362*7 + O(7^21)q^{7} + 212619578001532778 + O(7^21)q^{9} + -231380275661284448 + O(7^21)q^{11} + -31202403233731204 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{3} + 4q^{5} + 5q^{9} + 3q^{11} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 5q^{3} + 5q^{5} + 4q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -468q^{3} + 56214q^{5} + -47576*7q^{7} + -1375299q^{9} + 6397380q^{11} + 15199742q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{3} + q^{5} + 6q^{9} + q^{11} + 5q^{13} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
We have $\rho_f \otimes \chi^{5} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{3} + 3q^{5} + 3q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -19813083833991809 + O(7^20)q^{3} + -1619792884930289 + O(7^20)q^{5} + 1418566396767041*7 + O(7^20)q^{7} + -37912683339532212 + O(7^20)q^{9} + 11831635427128049 + O(7^20)q^{11} + -25421190565315715 + O(7^20)q^{13} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 4q^{3} + 5q^{5} + 4q^{7} + 3q^{9} + 2q^{11} + q^{13} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + 2q^{5} + 5q^{9} + q^{11} + \cdots \in S_{10}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{3} + 3q^{5} + 3q^{7} + 3q^{9} + 2q^{11} + 2q^{13} + \cdots \in S_{12}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{9} + 6q^{11} + \cdots \in S_{12}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 5q^{3} + 5q^{5} + 4q^{7} + 6q^{9} + 6q^{11} + 2q^{13} + \cdots \in S_{12}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{3} + 3q^{5} + 6q^{7} + q^{9} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{3} + 6q^{5} + 6q^{9} + 2q^{11} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{3} + 6q^{7} + q^{9} + 3q^{13} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{5} + q^{7} + 4q^{9} + 4q^{11} + 2q^{13} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{7}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\setcounter{section}{10}
\section{$\mathbf{2 \leq N \leq 16, \quad \ell = 11}$}

\subsection{$\mathbf{N = 2, \quad \ell = 11}$ \quad (8 forms)}

\begin{enumerate}
\item  Consider
$f = q + 9q^{2} + 4q^{3} + 4q^{4} + 3q^{5} + 3q^{6} + 3q^{7} + 3q^{8} + 5q^{10} + 5q^{12} + 5q^{13} + 5q^{14} + q^{15} + \cdots \in S_{14}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 9q^{3} + 3q^{4} + q^{5} + q^{6} + 5q^{7} + 4q^{8} + 5q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 64q^{2} + 1236q^{3} + 4096q^{4} + -57450q^{5} + 79104q^{6} + 64232q^{7} + 262144q^{8} + -6057*11q^{9} + -3676800q^{10} + 224052*11q^{11} + 5062656q^{12} + 8032766q^{13} + 4110848q^{14} + -71008200q^{15} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 4q^{3} + 5q^{4} + 2q^{5} + 5q^{6} + q^{7} + 9q^{8} + 4q^{9} + 8q^{10} + 9q^{12} + 2q^{13} + 4q^{14} + 8q^{15} + \cdots \in S_{16}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 3q^{2} + q^{3} + 9q^{4} + 10q^{5} + 3q^{6} + 4q^{7} + 5q^{8} + 3q^{9} + 8q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -128q^{2} + 6252q^{3} + 16384q^{4} + 90510q^{5} + -800256q^{6} + 56q^{7} + -2097152q^{8} + 24738597q^{9} + -11585280q^{10} + -8717268*11q^{11} + 102432768q^{12} + -59782138q^{13} + -7168q^{14} + 565868520q^{15} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 3q^{3} + 3q^{4} + 6q^{5} + 7q^{6} + 6q^{7} + 7q^{8} + 5q^{9} + 3q^{10} + 9q^{12} + 3q^{13} + 3q^{14} + 7q^{15} + \cdots \in S_{20}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 3q^{2} + q^{3} + 9q^{4} + 10q^{5} + 3q^{6} + 4q^{7} + 5q^{8} + 3q^{9} + 8q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{8}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 512q^{2} + -53028q^{3} + 262144q^{4} + -5556930q^{5} + -27150336q^{6} + -44496424q^{7} + 134217728q^{8} + 1649707317q^{9} + -2845148160q^{10} + 574606812*11q^{11} + -13900972032q^{12} + -33124973098q^{13} + -22782169088q^{14} + 294672884040q^{15} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 5q^{3} + q^{4} + 5q^{5} + 6q^{6} + 2q^{7} + 10q^{8} + 6q^{10} + 5q^{12} + 8q^{13} + 9q^{14} + 3q^{15} + \cdots \in S_{22}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 9q^{3} + 3q^{4} + q^{5} + q^{6} + 5q^{7} + 4q^{8} + 5q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{0}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1024q^{2} + 71604q^{3} + 1048576q^{4} + -28693770q^{5} + -73322496q^{6} + -853202392q^{7} + -1073741824q^{8} + -484838217*11q^{9} + 29382420480q^{10} + 7884652692*11q^{11} + 75082235904q^{12} + -895323442786q^{13} + 873679249408q^{14} + -2054588707080q^{15} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + q^{3} + 4q^{4} + 8q^{5} + 2q^{6} + q^{7} + 8q^{8} + 7q^{9} + 5q^{10} + 4q^{12} + 6q^{13} + 2q^{14} + 8q^{15} + \cdots \in S_{14}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 3q^{2} + q^{3} + 9q^{4} + 10q^{5} + 3q^{6} + 4q^{7} + 5q^{8} + 3q^{9} + 8q^{10} + 3q^{11} + 9q^{12} + 7q^{13} + q^{14} + 10q^{15} + \cdots \in S_{18}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 9q^{3} + 3q^{4} + q^{5} + q^{6} + 5q^{7} + 4q^{8} + 5q^{10} + 7q^{11} + 5q^{12} + 4q^{13} + 3q^{14} + 9q^{15} + \cdots \in S_{20}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 4q^{3} + q^{4} + 6q^{5} + 4q^{6} + 8q^{7} + q^{8} + 2q^{9} + 6q^{10} + 4q^{12} + 3q^{13} + 8q^{14} + 2q^{15} + \cdots \in S_{22}(\Gamma_0(2);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 3, \quad \ell = 11}$ \quad (13 forms)}

\begin{enumerate}
\item  Consider
$f = q + 10q^{2} + 8q^{3} + 4q^{4} + 7q^{5} + 3q^{6} + 7q^{7} + 4q^{8} + 9q^{9} + 4q^{10} + 10q^{12} + q^{13} + 4q^{14} + q^{15} + 8q^{16} + 8q^{17} + \cdots \in S_{14}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 8q^{2} + 7q^{3} + 3q^{4} + 6q^{5} + q^{6} + 8q^{7} + 9q^{8} + 5q^{9} + 4q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(2)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -12q^{2} + -729q^{3} + -8048q^{4} + -30210q^{5} + 8748q^{6} + 235088q^{7} + 194880q^{8} + 531441q^{9} + 362520q^{10} + -1016628*11q^{11} + 5866992q^{12} + 8049614q^{13} + -2821056q^{14} + 22023090q^{15} + 63590656q^{16} + -117494622q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 3q^{3} + 6q^{4} + 8q^{5} + 7q^{6} + q^{7} + 10q^{8} + 9q^{9} + 4q^{10} + 7q^{12} + 6q^{13} + 6q^{14} + 2q^{15} + q^{16} + 9q^{17} + \cdots \in S_{14}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{2} + 4q^{3} + 10q^{4} + 10q^{5} + 6q^{6} + 9q^{7} + 6q^{8} + 5q^{9} + 4q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 729q^{3} + -54*a + 800*11q^{4} + 128*a + 23814q^{5} + 729*aq^{6} + -3456*a - 103816q^{7} + 3524*a - 917568q^{8} + 531441q^{9} + 16902*a + 2174976q^{10} + -3328*11*a - 59292*11q^{11} + -39366*a + 583200*11q^{12} + 89856*a + 11192414q^{13} + 7528*11*a - 58724352q^{14} + 93312*a + 17360406q^{15} + -665496*a - 12209792q^{16} + 72960*a + 43889202q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 54x + -16992$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 2q^{3} + 10q^{4} + q^{5} + 5q^{6} + 6q^{7} + 4q^{9} + 8q^{10} + 9q^{12} + 8q^{13} + 4q^{14} + 2q^{15} + 10q^{16} + 10q^{17} + \cdots \in S_{16}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 6q^{2} + 6q^{3} + 7q^{4} + 5q^{5} + 3q^{6} + 2q^{7} + 3q^{9} + 8q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(9)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -234q^{2} + -2187q^{3} + 21988q^{4} + 280710q^{5} + 511758q^{6} + -1373344q^{7} + 229320*11q^{8} + 4782969q^{9} + -65686140q^{10} + 3093732*11q^{11} + -48087756q^{12} + 384022262q^{13} + 321362496q^{14} + -613912770q^{15} + -1310772464q^{16} + 1259207586q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 5q^{3} + 5q^{4} + 8q^{5} + 6q^{6} + 6q^{7} + 2q^{8} + 3q^{9} + 3q^{10} + 3q^{12} + 5q^{14} + 7q^{15} + 7q^{16} + q^{17} + \cdots \in S_{18}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 5q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{10} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{6}&0\\0&u(8)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 295774179405406427098 + O(11^20)q^{2} + 6561q^{3} + -1916136095384424821012 + O(11^21)q^{4} + 2310130633369952122661 + O(11^21)q^{5} + -309344301564058354907 + O(11^20)q^{6} + -96109659206662639477 + O(11^20)q^{7} + -48487063722068966619 + O(11^20)q^{8} + 43046721q^{9} + -30106657499102359755 + O(11^20)q^{10} + 24917664534207370931*11 + O(11^21)q^{11} + 1255733477402761297757 + O(11^21)q^{12} + -96133497373011434857*11 + O(11^21)q^{13} + -3004827105011682913199 + O(11^21)q^{14} + 1055199699543989498693 + O(11^21)q^{15} + 1852375435989639745917 + O(11^21)q^{16} + 233102293014980454352 + O(11^20)q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 5q^{3} + 5q^{4} + 8q^{5} + 5q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 8q^{10} + 3q^{12} + 5q^{14} + 7q^{15} + 7q^{16} + 10q^{17} + \cdots \in S_{18}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 5q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{10} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{1}&0\\0&u(8)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -295774179405406426504 + O(11^20)q^{2} + 6561q^{3} + 1916136095384424997528 + O(11^21)q^{4} + -2310130633369951739801 + O(11^21)q^{5} + 309344301564062252141 + O(11^20)q^{6} + 96109659206687111045 + O(11^20)q^{7} + 48487063722199306851 + O(11^20)q^{8} + 43046721q^{9} + 30106657498292977335 + O(11^20)q^{10} + -24917664534297148523*11 + O(11^21)q^{11} + -1255733477401603176281 + O(11^21)q^{12} + 96133497372782398653*11 + O(11^21)q^{13} + 3004827105011247347903 + O(11^21)q^{14} + -1055199699541477554233 + O(11^21)q^{15} + -1852375435939028421997 + O(11^21)q^{16} + -233102293049293580716 + O(11^20)q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 7q^{3} + 6q^{4} + 3q^{5} + 7q^{6} + 3q^{7} + 5q^{9} + 3q^{10} + 9q^{12} + q^{13} + 3q^{14} + 10q^{15} + 8q^{16} + 3q^{17} + \cdots \in S_{20}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 6q^{2} + 6q^{3} + 7q^{4} + 5q^{5} + 3q^{6} + 2q^{7} + 3q^{9} + 8q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{8}&0\\0&u(9)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -191290808516859524879 + O(11^20)q^{2} + -19683q^{3} + 263851407676615514982 + O(11^20)q^{4} + 93480575082868936053 + O(11^20)q^{5} + -204737600192343304640 + O(11^20)q^{6} + -225774540010305222868 + O(11^20)q^{7} + -7227616423302553478*11 + O(11^20)q^{8} + 387420489q^{9} + 230222509989842585614 + O(11^20)q^{10} + -4652872219213217838*11 + O(11^20)q^{11} + 242703580540089641014 + O(11^20)q^{12} + 201833677077681461484 + O(11^20)q^{13} + 318976276482900556072 + O(11^20)q^{14} + -6923215557643166464 + O(11^20)q^{15} + 33972979971878919738 + O(11^20)q^{16} + 134283332744169480116 + O(11^20)q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 10q^{3} + q^{4} + 8q^{5} + 6q^{6} + q^{7} + 6q^{8} + q^{9} + 7q^{10} + 10q^{12} + 6q^{13} + 5q^{14} + 3q^{15} + 6q^{16} + 5q^{17} + \cdots \in S_{22}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 8q^{2} + 7q^{3} + 3q^{4} + 6q^{5} + q^{6} + 8q^{7} + 9q^{8} + 5q^{9} + 4q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{0}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2844q^{2} + -59049q^{3} + 5991184q^{4} + 3109950q^{5} + 167935356q^{6} + 363303920q^{7} + -11074627008q^{8} + 3486784401q^{9} + -8844697800q^{10} + 1325621196*11q^{11} + -353773424016q^{12} + 113350790702q^{13} + -1033236348480q^{14} + -183639437550q^{15} + 18931815702784q^{16} + -8589389597982q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + q^{3} + 7q^{4} + 6q^{5} + 3q^{6} + 8q^{7} + 4q^{8} + q^{9} + 7q^{10} + 7q^{12} + 3q^{13} + 2q^{14} + 6q^{15} + 9q^{16} + 7q^{17} + \cdots \in S_{22}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{2} + 4q^{3} + 10q^{4} + 10q^{5} + 6q^{6} + 9q^{7} + 6q^{8} + 5q^{9} + 4q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 59049q^{3} + 666*a + 3040*11^2q^{4} + 13568*a - 4019706q^{5} + 59049*aq^{6} + -182016*a + 400559384q^{7} + -1285756*a + 1641684672q^{8} + 3486784401q^{9} + 5016582*a + 33445011456q^{10} + 2842112*11*a + 9047627748*11q^{11} + 39326634*a + 179508960*11^2q^{12} + -475236864*a + 134019420734q^{13} + 2308568*11^2*a - 448667983872q^{14} + 801176832*a - 237359619594q^{15} + -611332056*a - 3940794645632q^{16} + 1677304320*a - 6225306859278q^{17} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -666x + -2464992$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 5q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{10} + 8q^{11} + 3q^{12} + 9q^{14} + 10q^{15} + 8q^{16} + 2q^{17} + \cdots \in S_{16}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 6q^{3} + 7q^{4} + 5q^{5} + 3q^{6} + 2q^{7} + 3q^{9} + 8q^{10} + 9q^{11} + 9q^{12} + 6q^{13} + q^{14} + 8q^{15} + 6q^{16} + 6q^{17} + \cdots \in S_{18}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 4q^{3} + 10q^{4} + 10q^{5} + 6q^{6} + 9q^{7} + 6q^{8} + 5q^{9} + 4q^{10} + q^{11} + 7q^{12} + 7q^{13} + 8q^{14} + 7q^{15} + 4q^{16} + 3q^{17} + \cdots \in S_{20}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + 7q^{3} + 3q^{4} + 6q^{5} + q^{6} + 8q^{7} + 9q^{8} + 5q^{9} + 4q^{10} + 2q^{11} + 10q^{12} + 3q^{13} + 9q^{14} + 9q^{15} + 10q^{16} + 10q^{17} + \cdots \in S_{20}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 10q^{3} + 10q^{4} + 9q^{5} + 10q^{6} + 4q^{7} + 8q^{8} + q^{9} + 9q^{10} + q^{11} + q^{12} + 9q^{13} + 4q^{14} + 2q^{15} + 10q^{16} + 9q^{17} + \cdots \in S_{22}(\Gamma_0(3);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 4, \quad \ell = 11}$ \quad (6 forms)}

\begin{enumerate}
\item  Consider
$f = q + 6q^{3} + 4q^{5} + 7q^{7} + 9q^{9} + 8q^{13} + 2q^{15} + 2q^{17} + 3q^{19} + 9q^{21} + \cdots \in S_{14}(\Gamma_0(4);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 8q^{3} + 5q^{5} + 8q^{7} + 5q^{9} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(6)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 468q^{3} + 56214q^{5} + 333032q^{7} + -1375299q^{9} + -581580*11q^{11} + 15199742q^{13} + 26308152q^{15} + 43114194q^{17} + -365115484q^{19} + 155858976q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{3} + 6q^{5} + 4q^{15} + \cdots \in S_{18}(\Gamma_0(4);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 10q^{3} + 10q^{5} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{3} + -36*a + 196182q^{5} + 54*11*a + 1311040*11q^{7} + -5880*a + 207300141q^{9} + 315*11^2*a + 67443840*11q^{11} + -14796*11*a - 9409330*11^2q^{13} + 407862*a - 12111850944q^{15} + -15912*11*a - 1296700650*11q^{17} + -260145*11*a + 3832157056*11q^{19} + 90320*11^2*a + 18167776416*11q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 5880x + -336440304$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{3} + 3q^{5} + q^{7} + q^{9} + 4q^{13} + 6q^{15} + 4q^{17} + 10q^{19} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(4);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 8q^{3} + 5q^{5} + 8q^{7} + 5q^{9} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -197815801794667479828 + O(11^20)q^{3} + 10576129841434816530 + O(11^20)q^{5} + -186431864848526734843 + O(11^20)q^{7} + 118422391371438141362 + O(11^20)q^{9} + -24976012745428922040*11 + O(11^20)q^{11} + 298761524669349109867 + O(11^20)q^{13} + -310158248916494961548 + O(11^20)q^{15} + 66120954489554434749 + O(11^20)q^{17} + 39625584201932371499 + O(11^20)q^{19} + -3351172969707456781248 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 10q^{3} + 10q^{5} + q^{11} + q^{15} + \cdots \in S_{16}(\Gamma_0(4);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{3} + 5q^{5} + 8q^{7} + 5q^{9} + 6q^{11} + 2q^{13} + 7q^{15} + 8q^{17} + 4q^{19} + 9q^{21} + \cdots \in S_{20}(\Gamma_0(4);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 8q^{5} + 2q^{7} + 9q^{9} + 10q^{11} + 7q^{13} + 8q^{15} + 6q^{17} + 8q^{19} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(4);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 5, \quad \ell = 11}$ \quad (29 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 8q^{3} + 4q^{4} + 6q^{5} + 8q^{6} + 2q^{7} + 7q^{8} + 4q^{9} + 6q^{10} + 10q^{12} + 2q^{13} + 2q^{14} + 4q^{15} + 8q^{16} + 10q^{17} + 4q^{18} + 3q^{19} + 2q^{20} + 5q^{21} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 7q^{3} + 3q^{4} + 2q^{5} + 10q^{6} + 7q^{7} + 2q^{8} + q^{9} + 6q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -12620384221731986006 + O(11^20)q^{2} + 151444610660783831982 + O(11^20)q^{3} + -335869252126561139730 + O(11^20)q^{4} + -15625q^{5} + -215734353985867754612 + O(11^20)q^{6} + 208301256155830936705 + O(11^20)q^{7} + -313616457178931514788 + O(11^20)q^{8} + -277202792719173270143 + O(11^20)q^{9} + 77754949322198647857 + O(11^20)q^{10} + 467179384276462574*11 + O(11^20)q^{11} + -235359686522384348915 + O(11^20)q^{12} + -102850188749638604217 + O(11^20)q^{13} + 217376909472117945156 + O(11^20)q^{14} + -260309396933822358833 + O(11^20)q^{15} + -44070817469399131788 + O(11^20)q^{16} + 197645923551384171082 + O(11^20)q^{17} + 2766274810279464146062 + O(11^21)q^{18} + -93633860762009307517 + O(11^20)q^{19} + -165645991382823495751 + O(11^20)q^{20} + 172363870151720649110 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^63q^{2} + z^16q^{3} + z^57q^{4} + 5q^{5} + z^79q^{6} + z^75q^{7} + z^83q^{8} + z^73q^{9} + z^111q^{10} + z^73q^{12} + z^91q^{13} + z^18q^{14} + z^64q^{15} + z^78q^{16} + z^89q^{17} + z^16q^{18} + z^16q^{19} + z^105q^{20} + z^91q^{21} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^39q^{2} + z^64q^{3} + z^9q^{4} + 9q^{5} + z^103q^{6} + z^27q^{7} + z^11q^{8} + z^49q^{9} + z^111q^{10} + z^26q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^94)\omega^{1}&0\\0&u(z^26)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (53532297971390447749 + O(11^20))*a + 14587874043124250453 + O(11^20)q^{3} + -(321193787828342686396 + O(11^20))*a - 87527244258745500566 + O(11^20)q^{4} + 15625q^{5} + (2355421110741179700104 + O(11^21))*a + 641866457897467094284 + O(11^21)q^{6} + -(2100552445090505776236 + O(11^21))*a + 131290866388118275777*11 + O(11^21)q^{7} + (137481783789419152196 + O(11^20))*a - 319368775955780652738 + O(11^20)q^{8} + -(68795660465473189713 + O(11^20))*a + 333174423502429869670 + O(11^20)q^{9} + 15625*aq^{10} + (260468949276593413428*11 + O(11^21))*a - 235113773638832669599*11 + O(11^21)q^{11} + -(2038949325205391461800 + O(11^21))*a - 1888028016817827857410 + O(11^21)q^{12} + (26242871261719934174 + O(11^20))*a - 53383707803725881565 + O(11^20)q^{13} + -(327120698677696605894 + O(11^20))*a - 295435245628249738038 + O(11^20)q^{14} + (213912101803654641282 + O(11^20))*a - 126716358321429791014 + O(11^20)q^{15} + -(1403595937020214583060 + O(11^21))*a + 2786587190778092566760 + O(11^21)q^{16} + -(2495539722731842335796 + O(11^21))*a - 2606613883979187991749 + O(11^21)q^{17} + -(97111800611622346384 + O(11^20))*a + 99317520207522213159 + O(11^20)q^{18} + -(168466499355167487220 + O(11^20))*a - 243656292689409967035 + O(11^20)q^{19} + (62027379043193701960 + O(11^20))*a + 87548154996052361883 + O(11^20)q^{20} + (1494225993331619927446 + O(11^21))*a + 1936905577777111958077 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 321193787828342686396 + O(11^20)x + 87527244258745492374 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^93q^{2} + z^56q^{3} + z^27q^{4} + 5q^{5} + z^29q^{6} + z^105q^{7} + z^73q^{8} + z^83q^{9} + z^21q^{10} + z^83q^{12} + z^41q^{13} + z^78q^{14} + z^104q^{15} + z^18q^{16} + z^19q^{17} + z^56q^{18} + z^56q^{19} + z^75q^{20} + z^41q^{21} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^69q^{2} + z^104q^{3} + z^99q^{4} + 9q^{5} + z^53q^{6} + z^57q^{7} + zq^{8} + z^59q^{9} + z^21q^{10} + z^46q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^74)\omega^{1}&0\\0&u(z^46)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (53532297971390447749 + O(11^20))*a + 14587874043124250453 + O(11^20)q^{3} + -(321193787828342686396 + O(11^20))*a - 87527244258745500566 + O(11^20)q^{4} + 15625q^{5} + (2355421110741179700104 + O(11^21))*a + 641866457897467094284 + O(11^21)q^{6} + -(2100552445090505776236 + O(11^21))*a + 131290866388118275777*11 + O(11^21)q^{7} + (137481783789419152196 + O(11^20))*a - 319368775955780652738 + O(11^20)q^{8} + -(68795660465473189713 + O(11^20))*a + 333174423502429869670 + O(11^20)q^{9} + 15625*aq^{10} + (260468949276593413428*11 + O(11^21))*a - 235113773638832669599*11 + O(11^21)q^{11} + -(2038949325205391461800 + O(11^21))*a - 1888028016817827857410 + O(11^21)q^{12} + (26242871261719934174 + O(11^20))*a - 53383707803725881565 + O(11^20)q^{13} + -(327120698677696605894 + O(11^20))*a - 295435245628249738038 + O(11^20)q^{14} + (213912101803654641282 + O(11^20))*a - 126716358321429791014 + O(11^20)q^{15} + -(1403595937020214583060 + O(11^21))*a + 2786587190778092566760 + O(11^21)q^{16} + -(2495539722731842335796 + O(11^21))*a - 2606613883979187991749 + O(11^21)q^{17} + -(97111800611622346384 + O(11^20))*a + 99317520207522213159 + O(11^20)q^{18} + -(168466499355167487220 + O(11^20))*a - 243656292689409967035 + O(11^20)q^{19} + (62027379043193701960 + O(11^20))*a + 87548154996052361883 + O(11^20)q^{20} + (1494225993331619927446 + O(11^21))*a + 1936905577777111958077 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 321193787828342686396 + O(11^20)x + 87527244258745492374 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 9q^{6} + 7q^{7} + 5q^{8} + 2q^{9} + 10q^{10} + 3q^{12} + 5q^{13} + 5q^{14} + 7q^{15} + 8q^{16} + 3q^{18} + 3q^{19} + 7q^{20} + 9q^{21} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 7q^{3} + 2q^{4} + 4q^{5} + q^{6} + 6q^{7} + 4q^{8} + 7q^{9} + 10q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 26863872109196382498 + O(11^20)q^{2} + 53727744218392766176 + O(11^20)q^{3} + -254800414660158492240 + O(11^20)q^{4} + 78125q^{5} + 243268492633654005450 + O(11^20)q^{6} + 222357481521239421474 + O(11^20)q^{7} + -2636757158843895280 + O(11^20)q^{8} + -25974355622443124422 + O(11^20)q^{9} + -239975658619846050870 + O(11^20)q^{10} + 7210046684414903056*11 + O(11^20)q^{11} + 13470731334194788998 + O(11^20)q^{12} + -142311223279653905769 + O(11^20)q^{13} + 32194819596725715198 + O(11^20)q^{14} + 192798677692960094961 + O(11^20)q^{15} + 55022258457214287643 + O(11^20)q^{16} + 890737154567952352*11^2 + O(11^20)q^{17} + 127778294715179727358 + O(11^20)q^{18} + 25292277742610792565 + O(11^20)q^{19} + -282795265364094001611 + O(11^20)q^{20} + 73719810557565181326 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^69q^{2} + z^89q^{3} + z^32q^{4} + 8q^{5} + z^38q^{6} + z^58q^{7} + z^86q^{8} + z^27q^{9} + z^105q^{10} + zq^{12} + z^43q^{13} + z^7q^{14} + z^5q^{15} + z^106q^{16} + z^80q^{17} + 3q^{18} + z^88q^{19} + z^68q^{20} + z^27q^{21} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^21q^{2} + z^65q^{3} + z^56q^{4} + 7q^{5} + z^86q^{6} + z^82q^{7} + z^62q^{8} + z^99q^{9} + z^105q^{10} + z^113q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^7)\omega^{1}&0\\0&u(z^113)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (119332116899183789522 + O(11^20))*a - 289963247195975422809 + O(11^20)q^{3} + (33938867221558239249 + O(11^20))*a + 73927404995762355109 + O(11^20)q^{4} + -78125q^{5} + (101857504939825308548 + O(11^20))*a - 222769845645938376653 + O(11^20)q^{6} + (102745151811532260156 + O(11^20))*a + 229822133225343899379 + O(11^20)q^{7} + (135755468886232950796 + O(11^20))*a + 295709619983054959764 + O(11^20)q^{8} + (63030467333075780378 + O(11^20))*a + 312165537169705468046 + O(11^20)q^{9} + -78125*aq^{10} + -(168765785960933627012*11 + O(11^21))*a + 311152888191809111534*11 + O(11^21)q^{11} + (125443157808560691236 + O(11^20))*a - 87548066228650238690 + O(11^20)q^{12} + -(197858812923529688945 + O(11^20))*a - 131588734994259574436 + O(11^20)q^{13} + -(93063758791842907936 + O(11^20))*a + 93358383686554041597 + O(11^20)q^{14} + (147797026683051101208 + O(11^20))*a - 131892178513282872148 + O(11^20)q^{15} + -(33015971656111501483 + O(11^20))*a - 250301083578597829776 + O(11^20)q^{16} + (175182350764656660804 + O(11^20))*a - 203873197500330068637 + O(11^20)q^{17} + (29283455725163503173*11 + O(11^20))*a - 73711312622803942424 + O(11^20)q^{18} + -(248255047869677621248 + O(11^20))*a - 234685703463819140721 + O(11^20)q^{19} + -(166271655018445066984 + O(11^20))*a - 19808797906313900040 + O(11^20)q^{20} + (1577235821978196292390 + O(11^21))*a - 3681766448181318667329 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -33938867221558239249 + O(11^20)x + -73927404995762387877 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 6q^{3} + 10q^{4} + 8q^{5} + 7q^{6} + 9q^{7} + 2q^{9} + 2q^{10} + 5q^{12} + q^{13} + 5q^{14} + 4q^{15} + 10q^{16} + 2q^{17} + 6q^{18} + 8q^{19} + 3q^{20} + 10q^{21} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 7q^{2} + 2q^{3} + 8q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 10q^{9} + 9q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{4}&0\\0&u(10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -33938867221558239245 + O(11^20)q^{2} + 188105742256150117779 + O(11^20)q^{3} + -209682873881995318289 + O(11^20)q^{4} + -78125q^{5} + -154509316578261820091 + O(11^20)q^{6} + -136758374433501896649 + O(11^20)q^{7} + -28704830984146844140*11 + O(11^20)q^{8} + 38466444786103917783 + O(11^20)q^{9} + 166271655018444754484 + O(11^20)q^{10} + 49773419432827616160*11 + O(11^21)q^{11} + 98395801355221801117 + O(11^20)q^{12} + 71978677365746137636 + O(11^20)q^{13} + 101178225538479627081 + O(11^20)q^{14} + -210224454887110497731 + O(11^20)q^{15} + -305864306441435764892 + O(11^20)q^{16} + 161124223894243526842 + O(11^20)q^{17} + -203378932655237365306 + O(11^20)q^{18} + -307582526485900806407 + O(11^20)q^{19} + 12145423048017283775 + O(11^20)q^{20} + 188631173229543487486 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^39q^{2} + z^19q^{3} + z^112q^{4} + 8q^{5} + z^58q^{6} + z^38q^{7} + z^106q^{8} + z^57q^{9} + z^75q^{10} + z^11q^{12} + z^113q^{13} + z^77q^{14} + z^55q^{15} + z^86q^{16} + z^40q^{17} + 3q^{18} + z^8q^{19} + z^28q^{20} + z^57q^{21} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^111q^{2} + z^115q^{3} + z^16q^{4} + 7q^{5} + z^106q^{6} + z^62q^{7} + z^82q^{8} + z^9q^{9} + z^75q^{10} + z^43q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^77)\omega^{1}&0\\0&u(z^43)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (119332116899183789522 + O(11^20))*a - 289963247195975422809 + O(11^20)q^{3} + (33938867221558239249 + O(11^20))*a + 73927404995762355109 + O(11^20)q^{4} + -78125q^{5} + (101857504939825308548 + O(11^20))*a - 222769845645938376653 + O(11^20)q^{6} + (102745151811532260156 + O(11^20))*a + 229822133225343899379 + O(11^20)q^{7} + (135755468886232950796 + O(11^20))*a + 295709619983054959764 + O(11^20)q^{8} + (63030467333075780378 + O(11^20))*a + 312165537169705468046 + O(11^20)q^{9} + -78125*aq^{10} + -(168765785960933627012*11 + O(11^21))*a + 311152888191809111534*11 + O(11^21)q^{11} + (125443157808560691236 + O(11^20))*a - 87548066228650238690 + O(11^20)q^{12} + -(197858812923529688945 + O(11^20))*a - 131588734994259574436 + O(11^20)q^{13} + -(93063758791842907936 + O(11^20))*a + 93358383686554041597 + O(11^20)q^{14} + (147797026683051101208 + O(11^20))*a - 131892178513282872148 + O(11^20)q^{15} + -(33015971656111501483 + O(11^20))*a - 250301083578597829776 + O(11^20)q^{16} + (175182350764656660804 + O(11^20))*a - 203873197500330068637 + O(11^20)q^{17} + (29283455725163503173*11 + O(11^20))*a - 73711312622803942424 + O(11^20)q^{18} + -(248255047869677621248 + O(11^20))*a - 234685703463819140721 + O(11^20)q^{19} + -(166271655018445066984 + O(11^20))*a - 19808797906313900040 + O(11^20)q^{20} + (1577235821978196292390 + O(11^21))*a - 3681766448181318667329 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -33938867221558239249 + O(11^20)x + -73927404995762387877 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 10q^{3} + 9q^{4} + 4q^{5} + 4q^{6} + 9q^{7} + 3q^{8} + 3q^{9} + 6q^{10} + 2q^{12} + 8q^{14} + 7q^{15} + 2q^{16} + 3q^{17} + 10q^{18} + q^{19} + 3q^{20} + 2q^{21} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 7q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 5q^{7} + q^{8} + 4q^{9} + 6q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(6)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -41838085238737211681 + O(11^20)q^{2} + -125868660557771057862 + O(11^20)q^{3} + -116028672200071462526 + O(11^20)q^{4} + 390625q^{5} + -191689911494874837601 + O(11^20)q^{6} + 135217260697586253646 + O(11^20)q^{7} + 184115446319870555834 + O(11^20)q^{8} + -219923773394547572468 + O(11^20)q^{9} + 113580514956990629268 + O(11^20)q^{10} + -369373046933865619*11 + O(11^20)q^{11} + 84033159749155394923 + O(11^20)q^{12} + 1077580229623046632*11^2 + O(11^20)q^{13} + 317822314725039383938 + O(11^20)q^{14} + -184900728103764897866 + O(11^20)q^{15} + -41578797403087840344 + O(11^20)q^{16} + 276267507177063493545 + O(11^20)q^{17} + 329743213651131547816 + O(11^20)q^{18} + -180073874627271031835 + O(11^20)q^{19} + 139830448585330661821 + O(11^20)q^{20} + 42398176670751595751 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 10q^{3} + 9q^{4} + 4q^{5} + 7q^{6} + 2q^{7} + 8q^{8} + 3q^{9} + 5q^{10} + 2q^{12} + 8q^{14} + 7q^{15} + 2q^{16} + 8q^{17} + q^{18} + 10q^{19} + 3q^{20} + 9q^{21} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 7q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 5q^{7} + q^{8} + 4q^{9} + 6q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{6}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -52517281607646393853 + O(11^20)q^{2} + 313596772736366085416 + O(11^20)q^{3} + -63601593711270688482 + O(11^20)q^{4} + 390625q^{5} + 88084623947363806297 + O(11^20)q^{6} + -137852184341504600933 + O(11^20)q^{7} + 24751372135818747680 + O(11^20)q^{8} + 96587932484229413098 + O(11^20)q^{9} + 275217486612321747169 + O(11^20)q^{10} + -11097555990614619965*11 + O(11^20)q^{11} + -273444966375397611408 + O(11^20)q^{12} + 19880528258029213040*11 + O(11^20)q^{13} + -331339842428920666333 + O(11^20)q^{14} + -288977142052279757487 + O(11^20)q^{15} + 184699018207861145089 + O(11^20)q^{16} + 12968987894830027519 + O(11^20)q^{17} + 175467291678770841234 + O(11^20)q^{18} + -31065610046332008357 + O(11^20)q^{19} + 284769394328451511680 + O(11^20)q^{20} + 237673583155104514515 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 10q^{3} + 8q^{4} + 9q^{5} + 6q^{6} + 9q^{7} + 10q^{8} + 8q^{9} + q^{10} + 3q^{12} + 2q^{13} + q^{14} + 2q^{15} + 2q^{16} + 7q^{18} + 6q^{19} + 6q^{20} + 2q^{21} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 7q^{3} + 2q^{4} + 4q^{5} + q^{6} + 6q^{7} + 4q^{8} + 7q^{9} + 10q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{8}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -143279868018007443674 + O(11^20)q^{2} + -40999177994245764811 + O(11^20)q^{3} + -218618070134126456632 + O(11^20)q^{4} + 1953125q^{5} + -57975920002408264268 + O(11^20)q^{6} + 16788600972432273422 + O(11^20)q^{7} + 275074162722638796637 + O(11^20)q^{8} + -19902633163916402870 + O(11^20)q^{9} + 323169426198601558720 + O(11^20)q^{10} + -23704511305214108100*11 + O(11^20)q^{11} + 187764657299851644950 + O(11^20)q^{12} + -76209357487108445169 + O(11^20)q^{13} + -194874073895433378533 + O(11^20)q^{14} + 239626816425938701454 + O(11^20)q^{15} + -148088487879668702716 + O(11^20)q^{16} + 14199686208170960925*11 + O(11^20)q^{17} + 123182197907097793779 + O(11^20)q^{18} + 35027518924252861100 + O(11^20)q^{19} + -51196974290809583109 + O(11^20)q^{20} + -49579012172535498816 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 10q^{3} + 6q^{4} + 2q^{5} + q^{6} + 10q^{7} + 8q^{9} + 9q^{10} + 5q^{12} + 7q^{13} + q^{14} + 9q^{15} + 8q^{16} + 5q^{17} + 3q^{18} + 5q^{19} + q^{20} + q^{21} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 7q^{2} + 2q^{3} + 8q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 10q^{9} + 9q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(10)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -11419817576029337091 + O(11^20)q^{2} + -138376541028004409032 + O(11^20)q^{3} + 323426180412476008467 + O(11^20)q^{4} + -1953125q^{5} + -90270594417415877178 + O(11^20)q^{6} + 157757266900772466223 + O(11^20)q^{7} + 5208622346228947969*11 + O(11^20)q^{8} + -186723589698142585377 + O(11^20)q^{9} + -22128811795539190579 + O(11^20)q^{10} + 20440361386437411159*11 + O(11^20)q^{11} + 160731782944841853761 + O(11^20)q^{12} + -70264113356716138039 + O(11^20)q^{13} + 292942586750316565426 + O(11^20)q^{14} + 135231084048654270466 + O(11^20)q^{15} + -196056989340825550809 + O(11^20)q^{16} + 170872454165080198723 + O(11^20)q^{17} + 262717483627521215120 + O(11^20)q^{18} + -312701035815224350872 + O(11^20)q^{19} + 304123708667802353595 + O(11^20)q^{20} + -82318538399599896987 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^3q^{2} + z^91q^{3} + z^40q^{4} + 2q^{5} + z^94q^{6} + z^26q^{7} + z^118q^{8} + z^81q^{9} + z^15q^{10} + z^11q^{12} + z^77q^{13} + z^29q^{14} + z^103q^{15} + z^62q^{16} + z^76q^{17} + 7q^{18} + z^20q^{19} + z^52q^{20} + z^117q^{21} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^111q^{2} + z^115q^{3} + z^16q^{4} + 7q^{5} + z^106q^{6} + z^62q^{7} + z^82q^{8} + z^9q^{9} + z^75q^{10} + z^43q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^77)\omega^{8}&0\\0&u(z^43)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(272356213029084940251 + O(11^20))*a + 12992438387570805400 + O(11^20)q^{3} + -(282552299218640417025 + O(11^20))*a - 147420816882237526411 + O(11^20)q^{4} + -1953125q^{5} + (321349146368741906997 + O(11^20))*a + 199460733169605884512 + O(11^20)q^{6} + (281585915160954213610 + O(11^20))*a + 175450113922492945138 + O(11^20)q^{7} + -(296779586376047968571 + O(11^20))*a + 156351965950057937003 + O(11^20)q^{8} + (218611007891324290149 + O(11^20))*a + 56007533017502634031 + O(11^20)q^{9} + -1953125*aq^{10} + -(8054580602308956735*11 + O(11^20))*a - 12515340148482855933*11 + O(11^20)q^{11} + -(9262577459088346980 + O(11^20))*a + 88426330248628911181 + O(11^20)q^{12} + -(156323926578217706212 + O(11^20))*a + 64522289621963107047 + O(11^20)q^{13} + (268816343727391025066 + O(11^20))*a - 267669210612741564193 + O(11^20)q^{14} + (289329271526972476072 + O(11^20))*a + 273583131934250186720 + O(11^20)q^{15} + -(238570378441371333164 + O(11^20))*a + 17570412699848378530 + O(11^20)q^{16} + (123463712972720713231 + O(11^20))*a + 126277729342156318947 + O(11^20)q^{17} + (16670092660012589484*11 + O(11^20))*a - 62066153661114141103 + O(11^20)q^{18} + (153053537731965260672 + O(11^20))*a + 152088341463358382960 + O(11^20)q^{19} + -(225181746573845673180 + O(11^20))*a - 332858056054686470017 + O(11^20)q^{20} + (136494026686099183435 + O(11^20))*a - 228067542290854489548 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 282552299218640417025 + O(11^20)x + 147420816882237002123 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^33q^{2} + z^41q^{3} + z^80q^{4} + 2q^{5} + z^74q^{6} + z^46q^{7} + z^98q^{8} + z^51q^{9} + z^45q^{10} + zq^{12} + z^7q^{13} + z^79q^{14} + z^53q^{15} + z^82q^{16} + z^116q^{17} + 7q^{18} + z^100q^{19} + z^92q^{20} + z^87q^{21} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^21q^{2} + z^65q^{3} + z^56q^{4} + 7q^{5} + z^86q^{6} + z^82q^{7} + z^62q^{8} + z^99q^{9} + z^105q^{10} + z^113q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^7)\omega^{8}&0\\0&u(z^113)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(272356213029084940251 + O(11^20))*a + 12992438387570805400 + O(11^20)q^{3} + -(282552299218640417025 + O(11^20))*a - 147420816882237526411 + O(11^20)q^{4} + -1953125q^{5} + (321349146368741906997 + O(11^20))*a + 199460733169605884512 + O(11^20)q^{6} + (281585915160954213610 + O(11^20))*a + 175450113922492945138 + O(11^20)q^{7} + -(296779586376047968571 + O(11^20))*a + 156351965950057937003 + O(11^20)q^{8} + (218611007891324290149 + O(11^20))*a + 56007533017502634031 + O(11^20)q^{9} + -1953125*aq^{10} + -(8054580602308956735*11 + O(11^20))*a - 12515340148482855933*11 + O(11^20)q^{11} + -(9262577459088346980 + O(11^20))*a + 88426330248628911181 + O(11^20)q^{12} + -(156323926578217706212 + O(11^20))*a + 64522289621963107047 + O(11^20)q^{13} + (268816343727391025066 + O(11^20))*a - 267669210612741564193 + O(11^20)q^{14} + (289329271526972476072 + O(11^20))*a + 273583131934250186720 + O(11^20)q^{15} + -(238570378441371333164 + O(11^20))*a + 17570412699848378530 + O(11^20)q^{16} + (123463712972720713231 + O(11^20))*a + 126277729342156318947 + O(11^20)q^{17} + (16670092660012589484*11 + O(11^20))*a - 62066153661114141103 + O(11^20)q^{18} + (153053537731965260672 + O(11^20))*a + 152088341463358382960 + O(11^20)q^{19} + -(225181746573845673180 + O(11^20))*a - 332858056054686470017 + O(11^20)q^{20} + (136494026686099183435 + O(11^20))*a - 228067542290854489548 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 282552299218640417025 + O(11^20)x + 147420816882237002123 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 10q^{3} + q^{4} + 10q^{5} + 5q^{6} + 5q^{7} + 5q^{8} + 9q^{9} + 5q^{10} + 10q^{12} + q^{13} + 8q^{14} + q^{15} + 6q^{16} + 9q^{17} + 10q^{18} + 10q^{19} + 10q^{20} + 6q^{21} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 7q^{3} + 3q^{4} + 2q^{5} + 10q^{6} + 7q^{7} + 2q^{8} + q^{9} + 6q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 275144605214625748443 + O(11^20)q^{2} + 240721355629404215796 + O(11^20)q^{3} + -131467123738830936627 + O(11^20)q^{4} + -9765625q^{5} + 283213755417585405286 + O(11^20)q^{6} + 133978987453599872842 + O(11^20)q^{7} + -281812026498844332035 + O(11^20)q^{8} + -224440976202003808645 + O(11^20)q^{9} + -264788399425809942282 + O(11^20)q^{10} + 13025707697585856402*11 + O(11^20)q^{11} + 37097954646704543734 + O(11^20)q^{12} + 178663291681943713261 + O(11^20)q^{13} + -159334207672755924141 + O(11^20)q^{14} + -144775586509371702994 + O(11^20)q^{15} + -197323897962062355786 + O(11^20)q^{16} + -154795404961822146590 + O(11^20)q^{17} + 15993416421861507529 + O(11^20)q^{18} + 25608834240951565526 + O(11^20)q^{19} + 31432591590499097701 + O(11^20)q^{20} + 203305657421868531140 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^81q^{2} + z^80q^{3} + z^3q^{4} + q^{5} + z^41q^{6} + z^21q^{7} + z^37q^{8} + z^11q^{9} + z^81q^{10} + z^83q^{12} + z^29q^{13} + z^102q^{14} + z^80q^{15} + z^90q^{16} + z^31q^{17} + z^92q^{18} + z^20q^{19} + z^3q^{20} + z^101q^{21} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^69q^{2} + z^104q^{3} + z^99q^{4} + 9q^{5} + z^53q^{6} + z^57q^{7} + zq^{8} + z^59q^{9} + z^21q^{10} + z^46q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^74)\omega^{0}&0\\0&u(z^46)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(201253886205836665501 + O(11^20))*a + 282917300157211736178 + O(11^20)q^{3} + -(147063205062835800033 + O(11^20))*a - 118397767261058874098 + O(11^20)q^{4} + 9765625q^{5} + -(138770785959858363375 + O(11^20))*a - 80976933408913351870 + O(11^20)q^{6} + -(289621283629817684546 + O(11^20))*a + 1984080566642884678*11^2 + O(11^20)q^{7} + (270148319435878129057 + O(11^20))*a + 193812161616154491890 + O(11^20)q^{8} + (147557283009907406001 + O(11^20))*a - 227804422840343248626 + O(11^20)q^{9} + 9765625*aq^{10} + -(9722534301335260794*11 + O(11^20))*a - 3669875388785746732*11 + O(11^20)q^{11} + (48394546349081985981 + O(11^20))*a + 280878321188874515515 + O(11^20)q^{12} + -(249644661096362926151 + O(11^20))*a + 75358202062485679197 + O(11^20)q^{13} + -(76514543094535245394 + O(11^20))*a - 4676499495859155688 + O(11^20)q^{14} + -(165532877673333429328 + O(11^20))*a - 220341155100993587972 + O(11^20)q^{15} + -(238597242031777567192 + O(11^20))*a + 74558103200983251026 + O(11^20)q^{16} + (179995478818842509264 + O(11^20))*a - 247679175938471278396 + O(11^20)q^{17} + -(247138044754070174766 + O(11^20))*a - 74938610889099021873 + O(11^20)q^{18} + (58363019741151765335 + O(11^20))*a + 330052848558615776465 + O(11^20)q^{19} + -(296259554459095283659 + O(11^20))*a + 310382015518050109410 + O(11^20)q^{20} + (15286801533937813988 + O(11^20))*a - 180525604980275283537 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 147063205062835800033 + O(11^20)x + 118397767261056776946 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^51q^{2} + z^40q^{3} + z^33q^{4} + q^{5} + z^91q^{6} + z^111q^{7} + z^47q^{8} + zq^{9} + z^51q^{10} + z^73q^{12} + z^79q^{13} + z^42q^{14} + z^40q^{15} + z^30q^{16} + z^101q^{17} + z^52q^{18} + z^100q^{19} + z^33q^{20} + z^31q^{21} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^39q^{2} + z^64q^{3} + z^9q^{4} + 9q^{5} + z^103q^{6} + z^27q^{7} + z^11q^{8} + z^49q^{9} + z^111q^{10} + z^26q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^94)\omega^{0}&0\\0&u(z^26)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(201253886205836665501 + O(11^20))*a + 282917300157211736178 + O(11^20)q^{3} + -(147063205062835800033 + O(11^20))*a - 118397767261058874098 + O(11^20)q^{4} + 9765625q^{5} + -(138770785959858363375 + O(11^20))*a - 80976933408913351870 + O(11^20)q^{6} + -(289621283629817684546 + O(11^20))*a + 1984080566642884678*11^2 + O(11^20)q^{7} + (270148319435878129057 + O(11^20))*a + 193812161616154491890 + O(11^20)q^{8} + (147557283009907406001 + O(11^20))*a - 227804422840343248626 + O(11^20)q^{9} + 9765625*aq^{10} + -(9722534301335260794*11 + O(11^20))*a - 3669875388785746732*11 + O(11^20)q^{11} + (48394546349081985981 + O(11^20))*a + 280878321188874515515 + O(11^20)q^{12} + -(249644661096362926151 + O(11^20))*a + 75358202062485679197 + O(11^20)q^{13} + -(76514543094535245394 + O(11^20))*a - 4676499495859155688 + O(11^20)q^{14} + -(165532877673333429328 + O(11^20))*a - 220341155100993587972 + O(11^20)q^{15} + -(238597242031777567192 + O(11^20))*a + 74558103200983251026 + O(11^20)q^{16} + (179995478818842509264 + O(11^20))*a - 247679175938471278396 + O(11^20)q^{17} + -(247138044754070174766 + O(11^20))*a - 74938610889099021873 + O(11^20)q^{18} + (58363019741151765335 + O(11^20))*a + 330052848558615776465 + O(11^20)q^{19} + -(296259554459095283659 + O(11^20))*a + 310382015518050109410 + O(11^20)q^{20} + (15286801533937813988 + O(11^20))*a - 180525604980275283537 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 147063205062835800033 + O(11^20)x + 118397767261056776946 + O(11^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 7q^{2} + 2q^{3} + 8q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 10q^{9} + 9q^{10} + 10q^{11} + 5q^{12} + 6q^{13} + 9q^{14} + q^{15} + 2q^{16} + 4q^{17} + 4q^{18} + q^{19} + 4q^{20} + q^{21} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + q^{3} + 6q^{4} + 5q^{5} + 6q^{6} + q^{7} + 10q^{8} + 7q^{9} + 8q^{10} + 6q^{12} + 6q^{13} + 6q^{14} + 5q^{15} + q^{16} + 9q^{17} + 9q^{18} + 6q^{19} + 8q^{20} + q^{21} + \cdots \in S_{14}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{2} + 7q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 5q^{7} + q^{8} + 4q^{9} + 6q^{10} + 6q^{11} + 2q^{12} + 10q^{14} + 10q^{15} + 7q^{16} + 5q^{17} + 8q^{18} + 4q^{19} + 4q^{20} + 2q^{21} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^111q^{2} + z^115q^{3} + z^16q^{4} + 7q^{5} + z^106q^{6} + z^62q^{7} + z^82q^{8} + z^9q^{9} + z^75q^{10} + z^43q^{11} + z^11q^{12} + z^65q^{13} + z^53q^{14} + z^79q^{15} + z^14q^{16} + z^88q^{17} + q^{18} + z^104q^{19} + z^100q^{20} + z^57q^{21} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^21q^{2} + z^65q^{3} + z^56q^{4} + 7q^{5} + z^86q^{6} + z^82q^{7} + z^62q^{8} + z^99q^{9} + z^105q^{10} + z^113q^{11} + zq^{12} + z^115q^{13} + z^103q^{14} + z^29q^{15} + z^34q^{16} + z^8q^{17} + q^{18} + z^64q^{19} + z^20q^{20} + z^27q^{21} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + 7q^{3} + 2q^{4} + 4q^{5} + q^{6} + 6q^{7} + 4q^{8} + 7q^{9} + 10q^{10} + 7q^{11} + 3q^{12} + q^{13} + 4q^{14} + 6q^{15} + 7q^{16} + q^{18} + 9q^{19} + 8q^{20} + 9q^{21} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^69q^{2} + z^104q^{3} + z^99q^{4} + 9q^{5} + z^53q^{6} + z^57q^{7} + zq^{8} + z^59q^{9} + z^21q^{10} + z^46q^{11} + z^83q^{12} + z^17q^{13} + z^6q^{14} + z^56q^{15} + z^42q^{16} + z^43q^{17} + z^8q^{18} + z^104q^{19} + z^51q^{20} + z^41q^{21} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^39q^{2} + z^64q^{3} + z^9q^{4} + 9q^{5} + z^103q^{6} + z^27q^{7} + z^11q^{8} + z^49q^{9} + z^111q^{10} + z^26q^{11} + z^73q^{12} + z^67q^{13} + z^66q^{14} + z^16q^{15} + z^102q^{16} + z^113q^{17} + z^88q^{18} + z^64q^{19} + z^81q^{20} + z^91q^{21} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 7q^{3} + 3q^{4} + 2q^{5} + 10q^{6} + 7q^{7} + 2q^{8} + q^{9} + 6q^{10} + 3q^{11} + 10q^{12} + 6q^{13} + 10q^{14} + 3q^{15} + 10q^{16} + 7q^{17} + 3q^{18} + 4q^{19} + 6q^{20} + 5q^{21} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^5q^{2} + z^78q^{3} + z^32q^{4} + 10q^{5} + z^83q^{6} + 9q^{7} + z^57q^{8} + 5q^{9} + z^65q^{10} + q^{11} + z^110q^{12} + z^73q^{13} + z^77q^{14} + z^18q^{15} + 3q^{16} + z^23q^{17} + z^53q^{18} + z^92q^{20} + z^30q^{21} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^55q^{2} + z^18q^{3} + z^112q^{4} + 10q^{5} + z^73q^{6} + 9q^{7} + z^27q^{8} + 5q^{9} + z^115q^{10} + q^{11} + z^10q^{12} + z^83q^{13} + z^7q^{14} + z^78q^{15} + 3q^{16} + z^13q^{17} + z^103q^{18} + z^52q^{20} + z^90q^{21} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 7q^{4} + q^{5} + q^{6} + 8q^{7} + 4q^{8} + 2q^{9} + 3q^{10} + 6q^{12} + 3q^{13} + 2q^{14} + 4q^{15} + 9q^{16} + 7q^{17} + 6q^{18} + 9q^{19} + 7q^{20} + 10q^{21} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{2} + 10q^{4} + q^{5} + 8q^{8} + 8q^{9} + q^{10} + 10q^{11} + 2q^{13} + 10q^{16} + 6q^{17} + 8q^{18} + 7q^{19} + 10q^{20} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 6, \quad \ell = 11}$ \quad (13 forms)}

\begin{enumerate}
\item  Consider
$f = q + 7q^{2} + 9q^{3} + 5q^{4} + 8q^{5} + 8q^{6} + 2q^{7} + 2q^{8} + 4q^{9} + q^{10} + q^{12} + 9q^{13} + 3q^{14} + 6q^{15} + 3q^{16} + q^{17} + 6q^{18} + 7q^{20} + 7q^{21} + \cdots \in S_{16}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 5q^{3} + 9q^{4} + 7q^{5} + 7q^{6} + 8q^{7} + 6q^{8} + 3q^{9} + q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 128q^{2} + 2187q^{3} + 16384q^{4} + 77646q^{5} + 279936q^{6} + 762104q^{7} + 2097152q^{8} + 4782969q^{9} + 9938688q^{10} + 4364652*11q^{11} + 35831808q^{12} + 285130118q^{13} + 97549312q^{14} + 169811802q^{15} + 268435456q^{16} + -3173671566q^{17} + 612220032q^{18} + -535919660*11q^{19} + 1272152064q^{20} + 1666721448q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 2q^{3} + 5q^{4} + 8q^{5} + 3q^{6} + 9q^{7} + 2q^{8} + 4q^{9} + q^{10} + 10q^{12} + 10q^{13} + 8q^{14} + 5q^{15} + 3q^{16} + 3q^{17} + 6q^{18} + 6q^{19} + 7q^{20} + 7q^{21} + \cdots \in S_{16}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 9q^{2} + 8q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 6q^{7} + 3q^{8} + 9q^{9} + 10q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{4}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 128q^{2} + -2187q^{3} + 16384q^{4} + -114810q^{5} + -279936q^{6} + -3034528q^{7} + 2097152q^{8} + 4782969q^{9} + -14695680q^{10} + -9404700*11q^{11} + -35831808q^{12} + -104365834q^{13} + -388419584q^{14} + 251089470q^{15} + 268435456q^{16} + 997689762q^{17} + 612220032q^{18} + 4934015444q^{19} + -1881047040q^{20} + 6636512736q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 6q^{3} + 9q^{4} + 7q^{6} + 5q^{8} + 3q^{9} + 10q^{12} + 7q^{13} + 4q^{16} + 9q^{17} + 9q^{18} + 8q^{19} + \cdots \in S_{18}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 2q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 4q^{9} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(6)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 256q^{2} + -6561q^{3} + 65536q^{4} + -150*11^3q^{5} + -1679616q^{6} + 2269024*11q^{7} + 16777216q^{8} + 43046721q^{9} + -38400*11^3q^{10} + 11414220*11q^{11} + -429981696q^{12} + 4227195518q^{13} + 580870144*11q^{14} + 984150*11^3q^{15} + 4294967296q^{16} + 35551782594q^{17} + 11019960576q^{18} + -64354589764q^{19} + -9830400*11^3q^{20} + -14887066464*11q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 6q^{3} + 9q^{4} + 4q^{6} + 6q^{8} + 3q^{9} + 10q^{12} + 4q^{13} + 4q^{16} + 2q^{17} + 2q^{18} + 3q^{19} + \cdots \in S_{18}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 2q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 4q^{9} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{6}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -256q^{2} + -6561q^{3} + 65536q^{4} + 58650*11q^{5} + 1679616q^{6} + 361312*11q^{7} + -16777216q^{8} + 43046721q^{9} + -15014400*11q^{10} + -45460788*11q^{11} + -429981696q^{12} + -5425661314q^{13} + -92495872*11q^{14} + -384802650*11q^{15} + 4294967296q^{16} + -5466992958q^{17} + -11019960576q^{18} + -53889877060q^{19} + 3843686400*11q^{20} + -2370568032*11q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 7q^{3} + 3q^{4} + 2q^{5} + 2q^{6} + 10q^{7} + 4q^{8} + 5q^{9} + 10q^{10} + 10q^{12} + 4q^{13} + 6q^{14} + 3q^{15} + 9q^{16} + 2q^{17} + 3q^{18} + q^{19} + 6q^{20} + 4q^{21} + \cdots \in S_{20}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 9q^{2} + 8q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 6q^{7} + 3q^{8} + 9q^{9} + 10q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -512q^{2} + -19683q^{3} + 262144q^{4} + -3732474q^{5} + 10077696q^{6} + -149672656q^{7} + -134217728q^{8} + 387420489q^{9} + 1911026688q^{10} + -678152028*11q^{11} + -5159780352q^{12} + 59238459878q^{13} + 76632399872q^{14} + 73466285742q^{15} + 68719476736q^{16} + 523110429954q^{17} + -198359290368q^{18} + 969502037780q^{19} + -978445664256q^{20} + 2946006888048q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 4q^{3} + 3q^{4} + 2q^{5} + 9q^{6} + q^{7} + 4q^{8} + 5q^{9} + 10q^{10} + q^{12} + 8q^{13} + 5q^{14} + 8q^{15} + 9q^{16} + 8q^{17} + 3q^{18} + 6q^{20} + 4q^{21} + \cdots \in S_{20}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 5q^{3} + 9q^{4} + 7q^{5} + 7q^{6} + 8q^{7} + 6q^{8} + 3q^{9} + q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{8}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -512q^{2} + 19683q^{3} + 262144q^{4} + -5849490q^{5} + -10077696q^{6} + 173530952q^{7} + -134217728q^{8} + 387420489q^{9} + 2994938880q^{10} + -664802580*11q^{11} + 5159780352q^{12} + -41845065034q^{13} + -88847847424q^{14} + -115135511670q^{15} + 68719476736q^{16} + -95834399598q^{17} + -198359290368q^{18} + -219915683756*11q^{19} + -1533408706560q^{20} + 3415609728216q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 9q^{2} + 8q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 6q^{7} + 3q^{8} + 9q^{9} + 10q^{10} + q^{11} + 10q^{12} + 5q^{13} + 10q^{14} + 4q^{15} + 5q^{16} + 6q^{17} + 4q^{18} + 9q^{19} + 2q^{20} + 4q^{21} + \cdots \in S_{14}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 2q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 4q^{9} + 6q^{11} + 10q^{12} + 2q^{13} + 3q^{16} + 4q^{17} + 5q^{18} + 10q^{19} + \cdots \in S_{16}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + 5q^{3} + 9q^{4} + 7q^{5} + 7q^{6} + 8q^{7} + 6q^{8} + 3q^{9} + q^{10} + 6q^{11} + q^{12} + 4q^{13} + 9q^{14} + 2q^{15} + 4q^{16} + 5q^{17} + 2q^{18} + 8q^{20} + 7q^{21} + \cdots \in S_{18}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 4q^{3} + 3q^{4} + 10q^{5} + 2q^{6} + 9q^{7} + 7q^{8} + 5q^{9} + 5q^{10} + q^{11} + q^{12} + 7q^{13} + 10q^{14} + 7q^{15} + 9q^{16} + 3q^{17} + 8q^{18} + 8q^{19} + 8q^{20} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 10q^{3} + q^{4} + 2q^{5} + 10q^{6} + 7q^{7} + q^{8} + q^{9} + 2q^{10} + 10q^{11} + 10q^{12} + 5q^{13} + 7q^{14} + 9q^{15} + q^{16} + 2q^{17} + q^{18} + 4q^{19} + 2q^{20} + 4q^{21} + \cdots \in S_{22}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + 7q^{5} + q^{6} + 9q^{7} + q^{8} + q^{9} + 7q^{10} + q^{11} + q^{12} + 4q^{13} + 9q^{14} + 7q^{15} + q^{16} + 9q^{17} + q^{18} + 7q^{20} + 9q^{21} + \cdots \in S_{22}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + q^{3} + q^{4} + 10q^{6} + 2q^{7} + 10q^{8} + q^{9} + 10q^{11} + q^{12} + 7q^{13} + 9q^{14} + q^{16} + 5q^{17} + 10q^{18} + 7q^{19} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(6);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 7, \quad \ell = 11}$ \quad (43 forms)}

\begin{enumerate}
\item  Consider
$f = q + z^92q^{2} + 9q^{3} + z^81q^{4} + z^106q^{5} + z^44q^{6} + 7q^{7} + zq^{8} + 10q^{9} + z^78q^{10} + z^33q^{12} + z^79q^{13} + z^56q^{14} + z^58q^{15} + z^9q^{16} + z^44q^{17} + z^32q^{18} + z^45q^{19} + z^67q^{20} + 8q^{21} + \cdots \in S_{14}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^68q^{2} + q^{3} + z^33q^{4} + z^10q^{5} + z^68q^{6} + 8q^{7} + z^49q^{8} + 8q^{9} + z^78q^{10} + z^110q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^10)\omega^{1}&0\\0&u(z^110)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(27405027656081507142*11 + O(11^20))*a + 76192257249220610122 + O(11^20)q^{3} + (253713675939958896649 + O(11^20))*a - 241557092058087312783 + O(11^20)q^{4} + -(231615917718129726741 + O(11^20))*a + 228576771747661826352 + O(11^20)q^{5} + (14184447333117365038 + O(11^20))*a + 8413809707846420873*11 + O(11^20)q^{6} + -117649q^{7} + -(130944374886668774744 + O(11^20))*a - 225734439117230427277 + O(11^20)q^{8} + (18354764540037106737*11 + O(11^20))*a - 177113812277559718573 + O(11^20)q^{9} + -(324148185104364567748 + O(11^20))*a + 223271753433761017613 + O(11^20)q^{10} + -(32736093721667194711*11 + O(11^21))*a + 111753888953832168033*11 + O(11^21)q^{11} + -(329669831102922188292 + O(11^20))*a - 77279977617895811042 + O(11^20)q^{12} + (32129796828429361965 + O(11^20))*a - 107967458427797648454 + O(11^20)q^{13} + -117649*aq^{14} + (102588930153388913867 + O(11^20))*a + 259897339387239936865 + O(11^20)q^{15} + -(774779600808462352443 + O(11^21))*a - 201138724867539031094 + O(11^21)q^{16} + -(107497656441820267977 + O(11^20))*a + 5538118393333420979*11 + O(11^20)q^{17} + -(129745026733968502319 + O(11^20))*a - 12556083353382694743*11 + O(11^20)q^{18} + -(132369483296703034297 + O(11^20))*a - 317560857072964718418 + O(11^20)q^{19} + -(295202169116159997185 + O(11^20))*a - 122898259699358589179 + O(11^20)q^{20} + -(10831549184817619780*11 + O(11^20))*a - 221940632125997649054 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -253713675939958896649 + O(11^20)x + 241557092058087304591 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^52q^{2} + 9q^{3} + z^51q^{4} + z^86q^{5} + z^4q^{6} + 7q^{7} + z^11q^{8} + 10q^{9} + z^18q^{10} + z^3q^{12} + z^29q^{13} + z^16q^{14} + z^38q^{15} + z^99q^{16} + z^4q^{17} + z^112q^{18} + z^15q^{19} + z^17q^{20} + 8q^{21} + \cdots \in S_{14}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^28q^{2} + q^{3} + z^3q^{4} + z^110q^{5} + z^28q^{6} + 8q^{7} + z^59q^{8} + 8q^{9} + z^18q^{10} + z^10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^110)\omega^{1}&0\\0&u(z^10)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(27405027656081507142*11 + O(11^20))*a + 76192257249220610122 + O(11^20)q^{3} + (253713675939958896649 + O(11^20))*a - 241557092058087312783 + O(11^20)q^{4} + -(231615917718129726741 + O(11^20))*a + 228576771747661826352 + O(11^20)q^{5} + (14184447333117365038 + O(11^20))*a + 8413809707846420873*11 + O(11^20)q^{6} + -117649q^{7} + -(130944374886668774744 + O(11^20))*a - 225734439117230427277 + O(11^20)q^{8} + (18354764540037106737*11 + O(11^20))*a - 177113812277559718573 + O(11^20)q^{9} + -(324148185104364567748 + O(11^20))*a + 223271753433761017613 + O(11^20)q^{10} + -(32736093721667194711*11 + O(11^21))*a + 111753888953832168033*11 + O(11^21)q^{11} + -(329669831102922188292 + O(11^20))*a - 77279977617895811042 + O(11^20)q^{12} + (32129796828429361965 + O(11^20))*a - 107967458427797648454 + O(11^20)q^{13} + -117649*aq^{14} + (102588930153388913867 + O(11^20))*a + 259897339387239936865 + O(11^20)q^{15} + -(774779600808462352443 + O(11^21))*a - 201138724867539031094 + O(11^21)q^{16} + -(107497656441820267977 + O(11^20))*a + 5538118393333420979*11 + O(11^20)q^{17} + -(129745026733968502319 + O(11^20))*a - 12556083353382694743*11 + O(11^20)q^{18} + -(132369483296703034297 + O(11^20))*a - 317560857072964718418 + O(11^20)q^{19} + -(295202169116159997185 + O(11^20))*a - 122898259699358589179 + O(11^20)q^{20} + -(10831549184817619780*11 + O(11^20))*a - 221940632125997649054 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -253713675939958896649 + O(11^20)x + 241557092058087304591 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^850q^{2} + z^408q^{3} + z^552q^{4} + z^575q^{5} + z^1258q^{6} + 4q^{7} + z^1313q^{8} + z^496q^{9} + z^95q^{10} + z^960q^{12} + z^187q^{13} + z^1116q^{14} + z^983q^{15} + z^112q^{16} + z^32q^{17} + z^16q^{18} + z^965q^{19} + z^1127q^{20} + z^674q^{21} + \cdots \in S_{14}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^584q^{2} + z^940q^{3} + z^20q^{4} + z^841q^{5} + z^194q^{6} + 3q^{7} + z^515q^{8} + z^230q^{9} + z^95q^{10} + z^460q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^870)\omega^{1}&0\\0&u(z^460)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(73062537695835581310 + O(11^20))*a^2 + (20182199046657130027 + O(11^20))*a - 19085458489416885708*11 + O(11^20)q^{3} + a^2 - 8192q^{4} + (307437306453382102643 + O(11^20))*a^2 + (100910995233285650727 + O(11^20))*a + 295799772947191376720 + O(11^20)q^{5} + -(325497702777785185789 + O(11^20))*a^2 + (134568047065692216710 + O(11^20))*a - 185847300703495734115 + O(11^20)q^{6} + 117649q^{7} + -(82503212057148747198 + O(11^20))*a^2 - (322945264878481109917 + O(11^20))*a - 143463775974642050278 + O(11^20)q^{8} + (60338053741629420285 + O(11^20))*a^2 - (13745573973452655995 + O(11^20))*a - 304333433843260696454 + O(11^20)q^{9} + -(294712822499175941568 + O(11^20))*a^2 - (107347620306490141933 + O(11^20))*a + 218473704279657731649 + O(11^20)q^{10} + (20859038665989026243*11 + O(11^20))*a^2 - (2107930898399128340*11^2 + O(11^20))*a + 16734318217629676135*11 + O(11^20)q^{11} + (3025117775428787397912 + O(11^21))*a^2 - (2959173509638679583292 + O(11^21))*a + 2793588470984168460627 + O(11^21)q^{12} + (119443844231161382957 + O(11^20))*a^2 + (111468075218233833164 + O(11^20))*a + 19906582757566308358 + O(11^20)q^{13} + 117649*aq^{14} + (1156754406851822243*11 + O(11^20))*a^2 + (107437860702390453820 + O(11^20))*a + 17980889278896392910*11 + O(11^20)q^{15} + (19030612795030557951*11 + O(11^20))*a^2 - (26227782404289787462 + O(11^20))*a - 162978018279961939940 + O(11^20)q^{16} + (29140007820784670465 + O(11^20))*a^2 - (263391835261160920759 + O(11^20))*a + 8711997465235413189*11 + O(11^20)q^{17} + -(156376686294056584119 + O(11^20))*a^2 + (2411879672265064857*11 + O(11^20))*a - 59034091519339306820 + O(11^20)q^{18} + -(187579369185804595599 + O(11^20))*a^2 - (67253420233139422598 + O(11^20))*a - 15519192751975591322*11 + O(11^20)q^{19} + (264743144774869977495 + O(11^20))*a^2 + (91856352343470977542 + O(11^20))*a + 307210597158629948359 + O(11^20)q^{20} + -(7812124041067979013 + O(11^20))*a^2 + (280803523160418076194 + O(11^20))*a + 9740901684465867282*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 82503212057148747198 + O(11^20)x^{2} + 322945264878481093533 + O(11^20)x + 143463775974642050278 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^40q^{2} + z^498q^{3} + z^752q^{4} + z^1005q^{5} + z^538q^{6} + 4q^{7} + z^1143q^{8} + z^136q^{9} + z^1045q^{10} + z^1250q^{12} + z^727q^{13} + z^306q^{14} + z^173q^{15} + z^1232q^{16} + z^352q^{17} + z^176q^{18} + z^1305q^{19} + z^427q^{20} + z^764q^{21} + \cdots \in S_{14}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^1104q^{2} + z^1030q^{3} + z^220q^{4} + z^1271q^{5} + z^804q^{6} + 3q^{7} + z^345q^{8} + z^1200q^{9} + z^1045q^{10} + z^1070q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^260)\omega^{1}&0\\0&u(z^1070)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(73062537695835581310 + O(11^20))*a^2 + (20182199046657130027 + O(11^20))*a - 19085458489416885708*11 + O(11^20)q^{3} + a^2 - 8192q^{4} + (307437306453382102643 + O(11^20))*a^2 + (100910995233285650727 + O(11^20))*a + 295799772947191376720 + O(11^20)q^{5} + -(325497702777785185789 + O(11^20))*a^2 + (134568047065692216710 + O(11^20))*a - 185847300703495734115 + O(11^20)q^{6} + 117649q^{7} + -(82503212057148747198 + O(11^20))*a^2 - (322945264878481109917 + O(11^20))*a - 143463775974642050278 + O(11^20)q^{8} + (60338053741629420285 + O(11^20))*a^2 - (13745573973452655995 + O(11^20))*a - 304333433843260696454 + O(11^20)q^{9} + -(294712822499175941568 + O(11^20))*a^2 - (107347620306490141933 + O(11^20))*a + 218473704279657731649 + O(11^20)q^{10} + (20859038665989026243*11 + O(11^20))*a^2 - (2107930898399128340*11^2 + O(11^20))*a + 16734318217629676135*11 + O(11^20)q^{11} + (3025117775428787397912 + O(11^21))*a^2 - (2959173509638679583292 + O(11^21))*a + 2793588470984168460627 + O(11^21)q^{12} + (119443844231161382957 + O(11^20))*a^2 + (111468075218233833164 + O(11^20))*a + 19906582757566308358 + O(11^20)q^{13} + 117649*aq^{14} + (1156754406851822243*11 + O(11^20))*a^2 + (107437860702390453820 + O(11^20))*a + 17980889278896392910*11 + O(11^20)q^{15} + (19030612795030557951*11 + O(11^20))*a^2 - (26227782404289787462 + O(11^20))*a - 162978018279961939940 + O(11^20)q^{16} + (29140007820784670465 + O(11^20))*a^2 - (263391835261160920759 + O(11^20))*a + 8711997465235413189*11 + O(11^20)q^{17} + -(156376686294056584119 + O(11^20))*a^2 + (2411879672265064857*11 + O(11^20))*a - 59034091519339306820 + O(11^20)q^{18} + -(187579369185804595599 + O(11^20))*a^2 - (67253420233139422598 + O(11^20))*a - 15519192751975591322*11 + O(11^20)q^{19} + (264743144774869977495 + O(11^20))*a^2 + (91856352343470977542 + O(11^20))*a + 307210597158629948359 + O(11^20)q^{20} + -(7812124041067979013 + O(11^20))*a^2 + (280803523160418076194 + O(11^20))*a + 9740901684465867282*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 82503212057148747198 + O(11^20)x^{2} + 322945264878481093533 + O(11^20)x + 143463775974642050278 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^440q^{2} + z^158q^{3} + z^292q^{4} + z^415q^{5} + z^598q^{6} + 4q^{7} + z^603q^{8} + z^166q^{9} + z^855q^{10} + z^450q^{12} + z^17q^{13} + z^706q^{14} + z^573q^{15} + z^252q^{16} + z^1212q^{17} + z^606q^{18} + z^1055q^{19} + z^707q^{20} + z^424q^{21} + \cdots \in S_{14}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^174q^{2} + z^690q^{3} + z^1090q^{4} + z^681q^{5} + z^864q^{6} + 3q^{7} + z^1135q^{8} + z^1230q^{9} + z^855q^{10} + z^1130q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^200)\omega^{1}&0\\0&u(z^1130)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(73062537695835581310 + O(11^20))*a^2 + (20182199046657130027 + O(11^20))*a - 19085458489416885708*11 + O(11^20)q^{3} + a^2 - 8192q^{4} + (307437306453382102643 + O(11^20))*a^2 + (100910995233285650727 + O(11^20))*a + 295799772947191376720 + O(11^20)q^{5} + -(325497702777785185789 + O(11^20))*a^2 + (134568047065692216710 + O(11^20))*a - 185847300703495734115 + O(11^20)q^{6} + 117649q^{7} + -(82503212057148747198 + O(11^20))*a^2 - (322945264878481109917 + O(11^20))*a - 143463775974642050278 + O(11^20)q^{8} + (60338053741629420285 + O(11^20))*a^2 - (13745573973452655995 + O(11^20))*a - 304333433843260696454 + O(11^20)q^{9} + -(294712822499175941568 + O(11^20))*a^2 - (107347620306490141933 + O(11^20))*a + 218473704279657731649 + O(11^20)q^{10} + (20859038665989026243*11 + O(11^20))*a^2 - (2107930898399128340*11^2 + O(11^20))*a + 16734318217629676135*11 + O(11^20)q^{11} + (3025117775428787397912 + O(11^21))*a^2 - (2959173509638679583292 + O(11^21))*a + 2793588470984168460627 + O(11^21)q^{12} + (119443844231161382957 + O(11^20))*a^2 + (111468075218233833164 + O(11^20))*a + 19906582757566308358 + O(11^20)q^{13} + 117649*aq^{14} + (1156754406851822243*11 + O(11^20))*a^2 + (107437860702390453820 + O(11^20))*a + 17980889278896392910*11 + O(11^20)q^{15} + (19030612795030557951*11 + O(11^20))*a^2 - (26227782404289787462 + O(11^20))*a - 162978018279961939940 + O(11^20)q^{16} + (29140007820784670465 + O(11^20))*a^2 - (263391835261160920759 + O(11^20))*a + 8711997465235413189*11 + O(11^20)q^{17} + -(156376686294056584119 + O(11^20))*a^2 + (2411879672265064857*11 + O(11^20))*a - 59034091519339306820 + O(11^20)q^{18} + -(187579369185804595599 + O(11^20))*a^2 - (67253420233139422598 + O(11^20))*a - 15519192751975591322*11 + O(11^20)q^{19} + (264743144774869977495 + O(11^20))*a^2 + (91856352343470977542 + O(11^20))*a + 307210597158629948359 + O(11^20)q^{20} + -(7812124041067979013 + O(11^20))*a^2 + (280803523160418076194 + O(11^20))*a + 9740901684465867282*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 82503212057148747198 + O(11^20)x^{2} + 322945264878481093533 + O(11^20)x + 143463775974642050278 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 5q^{3} + 5q^{4} + 3q^{5} + q^{6} + 6q^{7} + 10q^{8} + 2q^{9} + 5q^{10} + 3q^{12} + q^{13} + 10q^{14} + 4q^{15} + 7q^{16} + 5q^{17} + 7q^{18} + 4q^{20} + 8q^{21} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 10q^{2} + 9q^{3} + 4q^{4} + 5q^{5} + 2q^{6} + 4q^{7} + 4q^{8} + 10q^{9} + 6q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{4}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 28722719146595434090 + O(11^20)q^{2} + 328822576687956964057 + O(11^20)q^{3} + 31478740502248114311 + O(11^20)q^{4} + 157196323713483771511 + O(11^20)q^{5} + 290698969731361434893 + O(11^20)q^{6} + 823543q^{7} + 93987852129612995916 + O(11^20)q^{8} + -301215984273656613203 + O(11^20)q^{9} + 102993401635051646823 + O(11^20)q^{10} + -8289955520052000907*11 + O(11^20)q^{11} + -183253795895146278794 + O(11^20)q^{12} + -325662347394605096303 + O(11^20)q^{13} + -168277679098906735491 + O(11^20)q^{14} + -300685747036274395693 + O(11^20)q^{15} + 101176278126498764735 + O(11^20)q^{16} + 317297279640967835951 + O(11^20)q^{17} + -303999164762280200327 + O(11^20)q^{18} + 1145298134780832843*11 + O(11^20)q^{19} + 188519264192668841920 + O(11^20)q^{20} + 166813106491786752225 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 8q^{3} + 10q^{4} + 3q^{5} + 2q^{6} + 6q^{7} + 8q^{9} + 9q^{10} + 3q^{12} + 2q^{13} + 7q^{14} + 2q^{15} + 10q^{16} + 10q^{17} + 2q^{18} + 6q^{19} + 8q^{20} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{2} + 2q^{3} + 7q^{4} + 4q^{5} + 10q^{6} + 2q^{7} + 6q^{9} + 9q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(9)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -138614148576207886975 + O(11^20)q^{2} + 239332666431501511602 + O(11^20)q^{3} + -121585747547483008593 + O(11^20)q^{4} + -335326179801247452914 + O(11^20)q^{5} + 277971151856717771124 + O(11^20)q^{6} + 823543q^{7} + 8459690774548121497*11 + O(11^20)q^{8} + -210837693083463794197 + O(11^20)q^{9} + 209301275982400929908 + O(11^20)q^{10} + 16528367446492011499*11 + O(11^20)q^{11} + 5081085829827769622 + O(11^20)q^{12} + 221497299944659678788 + O(11^20)q^{13} + 198379240540738210059 + O(11^20)q^{14} + 334474909313402187794 + O(11^20)q^{15} + 168787108538498063129 + O(11^20)q^{16} + -35601025975750472326 + O(11^20)q^{17} + 180708856832748527439 + O(11^20)q^{18} + 199129865421488667573 + O(11^20)q^{19} + 32513237723261157981 + O(11^20)q^{20} + -205904353517006444692 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^77q^{2} + z^5q^{3} + z^93q^{4} + z^102q^{5} + z^82q^{6} + 5q^{7} + 6q^{8} + z^17q^{9} + z^59q^{10} + z^98q^{12} + z^21q^{13} + z^5q^{14} + z^107q^{15} + z^51q^{16} + z^95q^{17} + z^94q^{18} + 5q^{19} + z^75q^{20} + z^53q^{21} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^29q^{2} + z^101q^{3} + z^117q^{4} + z^30q^{5} + z^10q^{6} + 9q^{7} + 7q^{8} + z^89q^{9} + z^59q^{10} + z^10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^110)\omega^{1}&0\\0&u(z^10)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(50463253003821933039 + O(11^20))*a + 220316573912364198*11^2 + O(11^20)q^{3} + (216077803275256990261 + O(11^20))*a + 261776061909737877606 + O(11^20)q^{4} + -(326676203302520720535 + O(11^20))*a + 83513661171343584101 + O(11^20)q^{5} + -(7234437342046056580 + O(11^20))*a - 175204820635375767623 + O(11^20)q^{6} + -823543q^{7} + (864428157559304036*11^2 + O(11^20))*a - 226914130103445350626 + O(11^20)q^{8} + (38385989983966513943 + O(11^20))*a - 22598052696525107668*11 + O(11^20)q^{9} + -(180636882297727091904 + O(11^20))*a - 231151081512365883589 + O(11^20)q^{10} + (6969265337691883746*11 + O(11^20))*a + 19861206531078616213*11 + O(11^20)q^{11} + (143592526238306783875 + O(11^20))*a - 104710372170636463212 + O(11^20)q^{12} + -(64713494215686064318 + O(11^20))*a + 174373707336937170362 + O(11^20)q^{13} + -823543*aq^{14} + (335052002151537678088 + O(11^20))*a + 275442710746747539592 + O(11^20)q^{15} + (162250831955272120249 + O(11^20))*a + 324199508623692953102 + O(11^20)q^{16} + -(235426534357400757006 + O(11^20))*a - 92968995082271008793 + O(11^20)q^{17} + -(107669562496056579982 + O(11^20))*a + 34554866955531112604 + O(11^20)q^{18} + -(13864543600948089143*11 + O(11^20))*a + 100026418944953616406 + O(11^20)q^{19} + -(251310441243754095833 + O(11^20))*a + 123480929389221313105 + O(11^20)q^{20} + (200581562564192354603 + O(11^20))*a + 2169378995333437440*11^2 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -216077803275256990261 + O(11^20)x + -261776061909737910374 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^7q^{2} + z^55q^{3} + z^63q^{4} + z^42q^{5} + z^62q^{6} + 5q^{7} + 6q^{8} + z^67q^{9} + z^49q^{10} + z^118q^{12} + z^111q^{13} + z^55q^{14} + z^97q^{15} + z^81q^{16} + z^85q^{17} + z^74q^{18} + 5q^{19} + z^105q^{20} + z^103q^{21} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^79q^{2} + z^31q^{3} + z^87q^{4} + z^90q^{5} + z^110q^{6} + 9q^{7} + 7q^{8} + z^19q^{9} + z^49q^{10} + z^110q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^10)\omega^{1}&0\\0&u(z^110)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(50463253003821933039 + O(11^20))*a + 220316573912364198*11^2 + O(11^20)q^{3} + (216077803275256990261 + O(11^20))*a + 261776061909737877606 + O(11^20)q^{4} + -(326676203302520720535 + O(11^20))*a + 83513661171343584101 + O(11^20)q^{5} + -(7234437342046056580 + O(11^20))*a - 175204820635375767623 + O(11^20)q^{6} + -823543q^{7} + (864428157559304036*11^2 + O(11^20))*a - 226914130103445350626 + O(11^20)q^{8} + (38385989983966513943 + O(11^20))*a - 22598052696525107668*11 + O(11^20)q^{9} + -(180636882297727091904 + O(11^20))*a - 231151081512365883589 + O(11^20)q^{10} + (6969265337691883746*11 + O(11^20))*a + 19861206531078616213*11 + O(11^20)q^{11} + (143592526238306783875 + O(11^20))*a - 104710372170636463212 + O(11^20)q^{12} + -(64713494215686064318 + O(11^20))*a + 174373707336937170362 + O(11^20)q^{13} + -823543*aq^{14} + (335052002151537678088 + O(11^20))*a + 275442710746747539592 + O(11^20)q^{15} + (162250831955272120249 + O(11^20))*a + 324199508623692953102 + O(11^20)q^{16} + -(235426534357400757006 + O(11^20))*a - 92968995082271008793 + O(11^20)q^{17} + -(107669562496056579982 + O(11^20))*a + 34554866955531112604 + O(11^20)q^{18} + -(13864543600948089143*11 + O(11^20))*a + 100026418944953616406 + O(11^20)q^{19} + -(251310441243754095833 + O(11^20))*a + 123480929389221313105 + O(11^20)q^{20} + (200581562564192354603 + O(11^20))*a + 2169378995333437440*11^2 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -216077803275256990261 + O(11^20)x + -261776061909737910374 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^61q^{2} + z^17q^{3} + z^21q^{4} + z^94q^{5} + z^78q^{6} + 2q^{7} + 8q^{8} + z^31q^{9} + z^35q^{10} + z^38q^{12} + z^46q^{13} + z^73q^{14} + z^111q^{15} + z^63q^{16} + z^94q^{17} + z^92q^{18} + z^105q^{19} + z^115q^{20} + z^29q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^49q^{2} + z^41q^{3} + z^117q^{4} + z^46q^{5} + z^90q^{6} + 5q^{7} + q^{8} + z^79q^{9} + z^95q^{10} + z^30q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^90)\omega^{6}&0\\0&u(z^30)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(83549459453171255905 + O(11^20))*a - 203315027988359294298 + O(11^20)q^{3} + (70364874143680768604 + O(11^20))*a + 321347652210027060895 + O(11^20)q^{4} + -(141487581123530331 + O(11^20))*a + 1563061403830260042 + O(11^20)q^{5} + (136168343641187347774 + O(11^20))*a + 11825848568837789669 + O(11^20)q^{6} + -5764801q^{7} + -(221008866026044798*11^3 + O(11^20))*a + 266948884840890161995 + O(11^20)q^{8} + (267533274523222212799 + O(11^20))*a - 62168635256483435100 + O(11^20)q^{9} + (319434250632006509729 + O(11^20))*a + 264335284056890048425 + O(11^20)q^{10} + (12582759249248148287*11 + O(11^20))*a - 5694512302716534655*11 + O(11^20)q^{11} + (41809132172570526897 + O(11^20))*a - 304096322625365345267 + O(11^20)q^{12} + (317100315838742281422 + O(11^20))*a + 193093552342265079022 + O(11^20)q^{13} + -5764801*aq^{14} + (330303627519030369318 + O(11^20))*a + 280456658996624934281 + O(11^20)q^{15} + -(141465265040415260480 + O(11^20))*a + 15090179863218615944 + O(11^20)q^{16} + -(273088038020356915738 + O(11^20))*a - 309778491326470733298 + O(11^20)q^{17} + -(205665703061335634504 + O(11^20))*a + 202214938946437019125 + O(11^20)q^{18} + -(231234716325588962311 + O(11^20))*a + 33315839557885297284 + O(11^20)q^{19} + (61369041587171186914 + O(11^20))*a + 322348265485277671542 + O(11^20)q^{20} + (67033063826465072769 + O(11^20))*a + 253490856139618348890 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -70364874143680768604 + O(11^20)x + -321347652210027191967 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^71q^{2} + z^67q^{3} + z^111q^{4} + z^74q^{5} + z^18q^{6} + 2q^{7} + 8q^{8} + z^101q^{9} + z^25q^{10} + z^58q^{12} + z^26q^{13} + z^83q^{14} + z^21q^{15} + z^93q^{16} + z^74q^{17} + z^52q^{18} + z^75q^{19} + z^65q^{20} + z^79q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^59q^{2} + z^91q^{3} + z^87q^{4} + z^26q^{5} + z^30q^{6} + 5q^{7} + q^{8} + z^29q^{9} + z^85q^{10} + z^90q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^30)\omega^{6}&0\\0&u(z^90)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(83549459453171255905 + O(11^20))*a - 203315027988359294298 + O(11^20)q^{3} + (70364874143680768604 + O(11^20))*a + 321347652210027060895 + O(11^20)q^{4} + -(141487581123530331 + O(11^20))*a + 1563061403830260042 + O(11^20)q^{5} + (136168343641187347774 + O(11^20))*a + 11825848568837789669 + O(11^20)q^{6} + -5764801q^{7} + -(221008866026044798*11^3 + O(11^20))*a + 266948884840890161995 + O(11^20)q^{8} + (267533274523222212799 + O(11^20))*a - 62168635256483435100 + O(11^20)q^{9} + (319434250632006509729 + O(11^20))*a + 264335284056890048425 + O(11^20)q^{10} + (12582759249248148287*11 + O(11^20))*a - 5694512302716534655*11 + O(11^20)q^{11} + (41809132172570526897 + O(11^20))*a - 304096322625365345267 + O(11^20)q^{12} + (317100315838742281422 + O(11^20))*a + 193093552342265079022 + O(11^20)q^{13} + -5764801*aq^{14} + (330303627519030369318 + O(11^20))*a + 280456658996624934281 + O(11^20)q^{15} + -(141465265040415260480 + O(11^20))*a + 15090179863218615944 + O(11^20)q^{16} + -(273088038020356915738 + O(11^20))*a - 309778491326470733298 + O(11^20)q^{17} + -(205665703061335634504 + O(11^20))*a + 202214938946437019125 + O(11^20)q^{18} + -(231234716325588962311 + O(11^20))*a + 33315839557885297284 + O(11^20)q^{19} + (61369041587171186914 + O(11^20))*a + 322348265485277671542 + O(11^20)q^{20} + (67033063826465072769 + O(11^20))*a + 253490856139618348890 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -70364874143680768604 + O(11^20)x + -321347652210027191967 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 2q^{3} + 8q^{4} + 6q^{5} + 7q^{6} + 2q^{7} + 9q^{8} + 6q^{9} + 10q^{10} + 5q^{12} + 5q^{13} + 7q^{14} + q^{15} + 3q^{16} + 3q^{17} + 10q^{18} + 3q^{19} + 4q^{20} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{2} + 8q^{3} + 2q^{4} + 10q^{5} + 8q^{6} + 6q^{7} + 3q^{8} + 8q^{9} + 10q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 24131187095258917320 + O(11^20)q^{2} + -79896737664886242989 + O(11^20)q^{3} + 46288417973476442046 + O(11^20)q^{4} + 85755590572600813497 + O(11^20)q^{5} + -36583156555439521361 + O(11^20)q^{6} + -5764801q^{7} + -132655106482342549514 + O(11^20)q^{8} + 303993345420337750942 + O(11^20)q^{9} + -288588688614718979180 + O(11^20)q^{10} + -879555530415950837*11 + O(11^20)q^{11} + -150762461044817885214 + O(11^20)q^{12} + -269483676632790469867 + O(11^20)q^{13} + -247545780943122670540 + O(11^20)q^{14} + 48221566315362850840 + O(11^20)q^{15} + 222144230469488194343 + O(11^20)q^{16} + -192004602532611321703 + O(11^20)q^{17} + 134054476507041136529 + O(11^20)q^{18} + 66671652416033454144 + O(11^20)q^{19} + 142999127060917988062 + O(11^20)q^{20} + -72343374275990105647 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + zq^{2} + z^17q^{3} + z^21q^{4} + z^94q^{5} + z^18q^{6} + 9q^{7} + 3q^{8} + z^31q^{9} + z^95q^{10} + z^38q^{12} + z^106q^{13} + z^73q^{14} + z^111q^{15} + z^63q^{16} + z^34q^{17} + z^32q^{18} + z^45q^{19} + z^115q^{20} + z^89q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^49q^{2} + z^41q^{3} + z^117q^{4} + z^46q^{5} + z^90q^{6} + 5q^{7} + q^{8} + z^79q^{9} + z^95q^{10} + z^30q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^90)\omega^{1}&0\\0&u(z^30)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(157598079783608786137 + O(11^20))*a + 155162673315769183015 + O(11^20)q^{3} + -(96164516882920856377 + O(11^20))*a + 51101753542662540511 + O(11^20)q^{4} + -(125411260536092387068 + O(11^20))*a + 287324528713869070836 + O(11^20)q^{5} + -(214080350725389220371 + O(11^20))*a + 102950832826561535050 + O(11^20)q^{6} + 5764801q^{7} + -(10546240007240075843*11 + O(11^20))*a + 63340971931890538702 + O(11^20)q^{8} + -(206871177026949503725 + O(11^20))*a + 230751259263087511561 + O(11^20)q^{9} + -(16294471694123876018 + O(11^20))*a - 140817166446489228602 + O(11^20)q^{10} + -(45293301924904850*11 + O(11^20))*a - 26509563813878221576*11 + O(11^20)q^{11} + (216185369467914843743 + O(11^20))*a + 99121465448923341474 + O(11^20)q^{12} + (64867884180449544749 + O(11^20))*a + 254306766073100318588 + O(11^20)q^{13} + 5764801*aq^{14} + (266201747921539143842 + O(11^20))*a - 318334688771765979196 + O(11^20)q^{15} + (235961190883791110746 + O(11^20))*a - 24200298886686916678 + O(11^20)q^{16} + -(164381232457909388501 + O(11^20))*a + 332689496964561037065 + O(11^20)q^{17} + (254841016044715280882 + O(11^20))*a - 176688589371178515395 + O(11^20)q^{18} + (276717176101804375470 + O(11^20))*a + 217499832924514295812 + O(11^20)q^{19} + (126601664666266708973 + O(11^20))*a - 146298366840380031243 + O(11^20)q^{20} + -(282527997656465790478 + O(11^20))*a + 268531037039380497425 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 96164516882920856377 + O(11^20)x + -51101753542662671583 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^11q^{2} + z^67q^{3} + z^111q^{4} + z^74q^{5} + z^78q^{6} + 9q^{7} + 3q^{8} + z^101q^{9} + z^85q^{10} + z^58q^{12} + z^86q^{13} + z^83q^{14} + z^21q^{15} + z^93q^{16} + z^14q^{17} + z^112q^{18} + z^15q^{19} + z^65q^{20} + z^19q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^59q^{2} + z^91q^{3} + z^87q^{4} + z^26q^{5} + z^30q^{6} + 5q^{7} + q^{8} + z^29q^{9} + z^85q^{10} + z^90q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^30)\omega^{1}&0\\0&u(z^90)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(157598079783608786137 + O(11^20))*a + 155162673315769183015 + O(11^20)q^{3} + -(96164516882920856377 + O(11^20))*a + 51101753542662540511 + O(11^20)q^{4} + -(125411260536092387068 + O(11^20))*a + 287324528713869070836 + O(11^20)q^{5} + -(214080350725389220371 + O(11^20))*a + 102950832826561535050 + O(11^20)q^{6} + 5764801q^{7} + -(10546240007240075843*11 + O(11^20))*a + 63340971931890538702 + O(11^20)q^{8} + -(206871177026949503725 + O(11^20))*a + 230751259263087511561 + O(11^20)q^{9} + -(16294471694123876018 + O(11^20))*a - 140817166446489228602 + O(11^20)q^{10} + -(45293301924904850*11 + O(11^20))*a - 26509563813878221576*11 + O(11^20)q^{11} + (216185369467914843743 + O(11^20))*a + 99121465448923341474 + O(11^20)q^{12} + (64867884180449544749 + O(11^20))*a + 254306766073100318588 + O(11^20)q^{13} + 5764801*aq^{14} + (266201747921539143842 + O(11^20))*a - 318334688771765979196 + O(11^20)q^{15} + (235961190883791110746 + O(11^20))*a - 24200298886686916678 + O(11^20)q^{16} + -(164381232457909388501 + O(11^20))*a + 332689496964561037065 + O(11^20)q^{17} + (254841016044715280882 + O(11^20))*a - 176688589371178515395 + O(11^20)q^{18} + (276717176101804375470 + O(11^20))*a + 217499832924514295812 + O(11^20)q^{19} + (126601664666266708973 + O(11^20))*a - 146298366840380031243 + O(11^20)q^{20} + -(282527997656465790478 + O(11^20))*a + 268531037039380497425 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 96164516882920856377 + O(11^20)x + -51101753542662671583 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 2q^{3} + 8q^{4} + 6q^{5} + 4q^{6} + 9q^{7} + 2q^{8} + 6q^{9} + q^{10} + 5q^{12} + 6q^{13} + 7q^{14} + q^{15} + 3q^{16} + 8q^{17} + q^{18} + 8q^{19} + 4q^{20} + 7q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{2} + 8q^{3} + 2q^{4} + 10q^{5} + 8q^{6} + 6q^{7} + 3q^{8} + 8q^{9} + 10q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{6}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -213723143574164042641 + O(11^20)q^{2} + 311769228536200123680 + O(11^20)q^{3} + -35861127409056105115 + O(11^20)q^{4} + -9959985508365214580 + O(11^20)q^{5} + 110518168045823772139 + O(11^20)q^{6} + 5764801q^{7} + 204522071995854136559 + O(11^20)q^{8} + 243723328102204421609 + O(11^20)q^{9} + -233714260169754892439 + O(11^20)q^{10} + 18449583309355997603*11 + O(11^20)q^{11} + 990016171475452258 + O(11^20)q^{12} + 99215669991482183574 + O(11^20)q^{13} + 257920026223429795155 + O(11^20)q^{14} + 18032029895007034083 + O(11^20)q^{15} + -313644292870144868904 + O(11^20)q^{16} + 107072577410306374743 + O(11^20)q^{17} + -287981081348848320905 + O(11^20)q^{18} + -82521241588880729319 + O(11^20)q^{19} + -42727239208129246030 + O(11^20)q^{20} + 293222649518929628527 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^91q^{2} + z^7q^{3} + z^111q^{4} + z^18q^{5} + z^98q^{6} + 8q^{7} + q^{8} + z^91q^{9} + z^109q^{10} + z^118q^{12} + z^75q^{13} + z^7q^{14} + z^25q^{15} + z^57q^{16} + zq^{17} + z^62q^{18} + 10q^{19} + z^9q^{20} + z^43q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^79q^{2} + z^31q^{3} + z^87q^{4} + z^90q^{5} + z^110q^{6} + 9q^{7} + 7q^{8} + z^19q^{9} + z^49q^{10} + z^110q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^10)\omega^{8}&0\\0&u(z^110)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(63650806289416642299 + O(11^20))*a - 13714945158597712626*11 + O(11^20)q^{3} + (47604408625681466157 + O(11^20))*a - 63963179489442255822 + O(11^20)q^{4} + (220962963433041813684 + O(11^20))*a + 134882930305930869812 + O(11^20)q^{5} + -(46821037972846985764 + O(11^20))*a - 202064088140581323629 + O(11^20)q^{6} + 40353607q^{7} + -(8558322368393945210*11 + O(11^20))*a - 189545800946950964922 + O(11^20)q^{8} + (84205600406603548027 + O(11^20))*a - 516468892227729230*11^2 + O(11^20)q^{9} + (28230145255149748150 + O(11^20))*a - 96959255885936012981 + O(11^20)q^{10} + (13230592897817609764*11 + O(11^20))*a + 18682018154584057522*11 + O(11^20)q^{11} + -(256252680459523242546 + O(11^20))*a + 120001759021678492263 + O(11^20)q^{12} + (24147729573368140423 + O(11^20))*a + 258214624012930926923 + O(11^20)q^{13} + 40353607*aq^{14} + -(118473110036973660985 + O(11^20))*a - 35088158028986371118 + O(11^20)q^{15} + -(271745214864993537257 + O(11^20))*a - 37917102360983444837 + O(11^20)q^{16} + -(218124924667851948619 + O(11^20))*a - 179436235600640133653 + O(11^20)q^{17} + -(295710641337425829004 + O(11^20))*a - 30867121417501933995 + O(11^20)q^{18} + -(14380888506915062089*11 + O(11^20))*a + 79095834150493342324 + O(11^20)q^{19} + -(64684527799523893104 + O(11^20))*a - 170499175204220164189 + O(11^20)q^{20} + -(324083581304264280523 + O(11^20))*a + 670047023073520937*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -47604408625681466157 + O(11^20)x + 63963179489441731534 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^41q^{2} + z^77q^{3} + z^21q^{4} + z^78q^{5} + z^118q^{6} + 8q^{7} + q^{8} + z^41q^{9} + z^119q^{10} + z^98q^{12} + z^105q^{13} + z^77q^{14} + z^35q^{15} + z^27q^{16} + z^11q^{17} + z^82q^{18} + 10q^{19} + z^99q^{20} + z^113q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^29q^{2} + z^101q^{3} + z^117q^{4} + z^30q^{5} + z^10q^{6} + 9q^{7} + 7q^{8} + z^89q^{9} + z^59q^{10} + z^10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^110)\omega^{8}&0\\0&u(z^10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(63650806289416642299 + O(11^20))*a - 13714945158597712626*11 + O(11^20)q^{3} + (47604408625681466157 + O(11^20))*a - 63963179489442255822 + O(11^20)q^{4} + (220962963433041813684 + O(11^20))*a + 134882930305930869812 + O(11^20)q^{5} + -(46821037972846985764 + O(11^20))*a - 202064088140581323629 + O(11^20)q^{6} + 40353607q^{7} + -(8558322368393945210*11 + O(11^20))*a - 189545800946950964922 + O(11^20)q^{8} + (84205600406603548027 + O(11^20))*a - 516468892227729230*11^2 + O(11^20)q^{9} + (28230145255149748150 + O(11^20))*a - 96959255885936012981 + O(11^20)q^{10} + (13230592897817609764*11 + O(11^20))*a + 18682018154584057522*11 + O(11^20)q^{11} + -(256252680459523242546 + O(11^20))*a + 120001759021678492263 + O(11^20)q^{12} + (24147729573368140423 + O(11^20))*a + 258214624012930926923 + O(11^20)q^{13} + 40353607*aq^{14} + -(118473110036973660985 + O(11^20))*a - 35088158028986371118 + O(11^20)q^{15} + -(271745214864993537257 + O(11^20))*a - 37917102360983444837 + O(11^20)q^{16} + -(218124924667851948619 + O(11^20))*a - 179436235600640133653 + O(11^20)q^{17} + -(295710641337425829004 + O(11^20))*a - 30867121417501933995 + O(11^20)q^{18} + -(14380888506915062089*11 + O(11^20))*a + 79095834150493342324 + O(11^20)q^{19} + -(64684527799523893104 + O(11^20))*a - 170499175204220164189 + O(11^20)q^{20} + -(324083581304264280523 + O(11^20))*a + 670047023073520937*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -47604408625681466157 + O(11^20)x + 63963179489441731534 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 6q^{3} + 6q^{4} + 9q^{5} + 5q^{6} + 3q^{7} + 10q^{9} + 2q^{10} + 3q^{12} + 3q^{13} + 8q^{14} + 10q^{15} + 8q^{16} + 3q^{17} + q^{18} + q^{19} + 10q^{20} + 7q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{2} + 2q^{3} + 7q^{4} + 4q^{5} + 10q^{6} + 2q^{7} + 6q^{9} + 9q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{8}&0\\0&u(9)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -231211139539057666297 + O(11^20)q^{2} + -77034244987525121851 + O(11^20)q^{3} + -58200419591501482861 + O(11^20)q^{4} + 278354185510559692939 + O(11^20)q^{5} + -321189171921997818412 + O(11^20)q^{6} + -40353607q^{7} + 14030118723786888339*11 + O(11^20)q^{8} + 36761587830696740153 + O(11^20)q^{9} + 28764661846920058820 + O(11^20)q^{10} + 4428561558230320354*11 + O(11^20)q^{11} + -80172733265751448825 + O(11^20)q^{12} + -283854260540937371194 + O(11^20)q^{13} + -51489632209200113074 + O(11^20)q^{14} + 171827902724384092999 + O(11^20)q^{15} + 53447487331063155085 + O(11^20)q^{16} + -146131003072360743687 + O(11^20)q^{17} + -11697569339195017805 + O(11^20)q^{18} + -86517437311746740056 + O(11^20)q^{19} + -259386062405783112801 + O(11^20)q^{20} + 108683682007286845807 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + q^{3} + 3q^{4} + 9q^{5} + 8q^{6} + 3q^{7} + 9q^{8} + 8q^{9} + 6q^{10} + 3q^{12} + 7q^{13} + 2q^{14} + 9q^{15} + 10q^{16} + 7q^{17} + 9q^{18} + 5q^{20} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 10q^{2} + 9q^{3} + 4q^{4} + 5q^{5} + 2q^{6} + 4q^{7} + 4q^{8} + 10q^{9} + 6q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 159688429475667623000 + O(11^20)q^{2} + -120566352222240894507 + O(11^20)q^{3} + 294997228400355812620 + O(11^20)q^{4} + 216973890736726035350 + O(11^20)q^{5} + -124877248115583828351 + O(11^20)q^{6} + -40353607q^{7} + -2239765413685705193 + O(11^20)q^{8} + 166491426698578621351 + O(11^20)q^{9} + -322623372507349228276 + O(11^20)q^{10} + 10058491843486235771*11 + O(11^20)q^{11} + 276162620853913445402 + O(11^20)q^{12} + 36448223358161613819 + O(11^20)q^{13} + 321452701848086556002 + O(11^20)q^{14} + -252962986110195526882 + O(11^20)q^{15} + -218780709015277011494 + O(11^20)q^{16} + 10646878445570297608 + O(11^20)q^{17} + -326382961113593897878 + O(11^20)q^{18} + 22824646337837639986*11 + O(11^20)q^{19} + 295682211349000052549 + O(11^20)q^{20} + 269397302615882866010 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^717q^{2} + z^674q^{3} + z^286q^{4} + z^43q^{5} + z^61q^{6} + 10q^{7} + z^914q^{8} + z^1028q^{9} + z^760q^{10} + z^960q^{12} + z^54q^{13} + z^52q^{14} + z^717q^{15} + z^910q^{16} + z^165q^{17} + z^415q^{18} + z^566q^{19} + z^329q^{20} + z^9q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^584q^{2} + z^940q^{3} + z^20q^{4} + z^841q^{5} + z^194q^{6} + 3q^{7} + z^515q^{8} + z^230q^{9} + z^95q^{10} + z^460q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^870)\omega^{0}&0\\0&u(z^460)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (31661743643353139695 + O(11^20))*a^2 + (316000893364366672092 + O(11^20))*a + 15972496321807558330*11 + O(11^20)q^{3} + a^2 - 2097152q^{4} + (140132178330796586431 + O(11^20))*a^2 - (295145357767755056168 + O(11^20))*a - 8348824864292780635 + O(11^20)q^{5} + -(28962804087306400823 + O(11^20))*a^2 - (105209807630771820175 + O(11^20))*a - 218862079324714696825 + O(11^20)q^{6} + -282475249q^{7} + (53589855169818165317 + O(11^20))*a^2 + (219600516926201824268 + O(11^20))*a + 6817775513041859439 + O(11^20)q^{8} + -(186951769925361874028 + O(11^20))*a^2 - (278045242069704139535 + O(11^20))*a - 278816808866691125620 + O(11^20)q^{9} + -(110953986250322069784 + O(11^20))*a^2 + (9081258750047912291 + O(11^20))*a + 164788606814319855579 + O(11^20)q^{10} + (15471912630176697369*11 + O(11^20))*a^2 - (23701510831976800056*11 + O(11^20))*a - 23327100012336069053*11 + O(11^20)q^{11} + (306473781009761053261 + O(11^20))*a^2 + (215294955741384913794 + O(11^20))*a + 211831800795252434128 + O(11^20)q^{12} + -(144592111549391845291 + O(11^20))*a^2 - (77341095632517178760 + O(11^20))*a + 223310345751536819058 + O(11^20)q^{13} + -282475249*aq^{14} + -(12386429415790948611*11 + O(11^20))*a^2 + (242885730649114384437 + O(11^20))*a - 30073564064299869065*11 + O(11^20)q^{15} + (931841605152376*11^5 + O(11^20))*a^2 + (296695343657291870327 + O(11^20))*a - 157133299523256026332 + O(11^20)q^{16} + (22618431954197105555 + O(11^20))*a^2 - (11526264815621578944 + O(11^20))*a + 15668050800309374568*11 + O(11^20)q^{17} + -(319170368555686317555 + O(11^20))*a^2 - (2309231249489196357*11^2 + O(11^20))*a - 213824047714356489644 + O(11^20)q^{18} + (335398715144540592278 + O(11^20))*a^2 + (311790150275739254953 + O(11^20))*a - 19525242245031416232*11 + O(11^20)q^{19} + -(50602760664477573252 + O(11^20))*a^2 - (287002380627607310275 + O(11^20))*a - 245582865097477214232 + O(11^20)q^{20} + -(64547799639717168076 + O(11^20))*a^2 + (29315361939286459208 + O(11^20))*a + 27769851271572784676*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -53589855169818165317 + O(11^20)x^{2} + -219600516926206018572 + O(11^20)x + -6817775513041859439 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^1237q^{2} + z^764q^{3} + z^486q^{4} + z^473q^{5} + z^671q^{6} + 10q^{7} + z^744q^{8} + z^668q^{9} + z^380q^{10} + z^1250q^{12} + z^594q^{13} + z^572q^{14} + z^1237q^{15} + z^700q^{16} + z^485q^{17} + z^575q^{18} + z^906q^{19} + z^959q^{20} + z^99q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^1104q^{2} + z^1030q^{3} + z^220q^{4} + z^1271q^{5} + z^804q^{6} + 3q^{7} + z^345q^{8} + z^1200q^{9} + z^1045q^{10} + z^1070q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^260)\omega^{0}&0\\0&u(z^1070)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (31661743643353139695 + O(11^20))*a^2 + (316000893364366672092 + O(11^20))*a + 15972496321807558330*11 + O(11^20)q^{3} + a^2 - 2097152q^{4} + (140132178330796586431 + O(11^20))*a^2 - (295145357767755056168 + O(11^20))*a - 8348824864292780635 + O(11^20)q^{5} + -(28962804087306400823 + O(11^20))*a^2 - (105209807630771820175 + O(11^20))*a - 218862079324714696825 + O(11^20)q^{6} + -282475249q^{7} + (53589855169818165317 + O(11^20))*a^2 + (219600516926201824268 + O(11^20))*a + 6817775513041859439 + O(11^20)q^{8} + -(186951769925361874028 + O(11^20))*a^2 - (278045242069704139535 + O(11^20))*a - 278816808866691125620 + O(11^20)q^{9} + -(110953986250322069784 + O(11^20))*a^2 + (9081258750047912291 + O(11^20))*a + 164788606814319855579 + O(11^20)q^{10} + (15471912630176697369*11 + O(11^20))*a^2 - (23701510831976800056*11 + O(11^20))*a - 23327100012336069053*11 + O(11^20)q^{11} + (306473781009761053261 + O(11^20))*a^2 + (215294955741384913794 + O(11^20))*a + 211831800795252434128 + O(11^20)q^{12} + -(144592111549391845291 + O(11^20))*a^2 - (77341095632517178760 + O(11^20))*a + 223310345751536819058 + O(11^20)q^{13} + -282475249*aq^{14} + -(12386429415790948611*11 + O(11^20))*a^2 + (242885730649114384437 + O(11^20))*a - 30073564064299869065*11 + O(11^20)q^{15} + (931841605152376*11^5 + O(11^20))*a^2 + (296695343657291870327 + O(11^20))*a - 157133299523256026332 + O(11^20)q^{16} + (22618431954197105555 + O(11^20))*a^2 - (11526264815621578944 + O(11^20))*a + 15668050800309374568*11 + O(11^20)q^{17} + -(319170368555686317555 + O(11^20))*a^2 - (2309231249489196357*11^2 + O(11^20))*a - 213824047714356489644 + O(11^20)q^{18} + (335398715144540592278 + O(11^20))*a^2 + (311790150275739254953 + O(11^20))*a - 19525242245031416232*11 + O(11^20)q^{19} + -(50602760664477573252 + O(11^20))*a^2 - (287002380627607310275 + O(11^20))*a - 245582865097477214232 + O(11^20)q^{20} + -(64547799639717168076 + O(11^20))*a^2 + (29315361939286459208 + O(11^20))*a + 27769851271572784676*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -53589855169818165317 + O(11^20)x^{2} + -219600516926206018572 + O(11^20)x + -6817775513041859439 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^307q^{2} + z^424q^{3} + z^26q^{4} + z^1213q^{5} + z^731q^{6} + 10q^{7} + z^204q^{8} + z^698q^{9} + z^190q^{10} + z^450q^{12} + z^1214q^{13} + z^972q^{14} + z^307q^{15} + z^1050q^{16} + z^15q^{17} + z^1005q^{18} + z^656q^{19} + z^1239q^{20} + z^1089q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^174q^{2} + z^690q^{3} + z^1090q^{4} + z^681q^{5} + z^864q^{6} + 3q^{7} + z^1135q^{8} + z^1230q^{9} + z^855q^{10} + z^1130q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^200)\omega^{0}&0\\0&u(z^1130)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (31661743643353139695 + O(11^20))*a^2 + (316000893364366672092 + O(11^20))*a + 15972496321807558330*11 + O(11^20)q^{3} + a^2 - 2097152q^{4} + (140132178330796586431 + O(11^20))*a^2 - (295145357767755056168 + O(11^20))*a - 8348824864292780635 + O(11^20)q^{5} + -(28962804087306400823 + O(11^20))*a^2 - (105209807630771820175 + O(11^20))*a - 218862079324714696825 + O(11^20)q^{6} + -282475249q^{7} + (53589855169818165317 + O(11^20))*a^2 + (219600516926201824268 + O(11^20))*a + 6817775513041859439 + O(11^20)q^{8} + -(186951769925361874028 + O(11^20))*a^2 - (278045242069704139535 + O(11^20))*a - 278816808866691125620 + O(11^20)q^{9} + -(110953986250322069784 + O(11^20))*a^2 + (9081258750047912291 + O(11^20))*a + 164788606814319855579 + O(11^20)q^{10} + (15471912630176697369*11 + O(11^20))*a^2 - (23701510831976800056*11 + O(11^20))*a - 23327100012336069053*11 + O(11^20)q^{11} + (306473781009761053261 + O(11^20))*a^2 + (215294955741384913794 + O(11^20))*a + 211831800795252434128 + O(11^20)q^{12} + -(144592111549391845291 + O(11^20))*a^2 - (77341095632517178760 + O(11^20))*a + 223310345751536819058 + O(11^20)q^{13} + -282475249*aq^{14} + -(12386429415790948611*11 + O(11^20))*a^2 + (242885730649114384437 + O(11^20))*a - 30073564064299869065*11 + O(11^20)q^{15} + (931841605152376*11^5 + O(11^20))*a^2 + (296695343657291870327 + O(11^20))*a - 157133299523256026332 + O(11^20)q^{16} + (22618431954197105555 + O(11^20))*a^2 - (11526264815621578944 + O(11^20))*a + 15668050800309374568*11 + O(11^20)q^{17} + -(319170368555686317555 + O(11^20))*a^2 - (2309231249489196357*11^2 + O(11^20))*a - 213824047714356489644 + O(11^20)q^{18} + (335398715144540592278 + O(11^20))*a^2 + (311790150275739254953 + O(11^20))*a - 19525242245031416232*11 + O(11^20)q^{19} + -(50602760664477573252 + O(11^20))*a^2 - (287002380627607310275 + O(11^20))*a - 245582865097477214232 + O(11^20)q^{20} + -(64547799639717168076 + O(11^20))*a^2 + (29315361939286459208 + O(11^20))*a + 27769851271572784676*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -53589855169818165317 + O(11^20)x^{2} + -219600516926206018572 + O(11^20)x + -6817775513041859439 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^80q^{2} + 3q^{3} + z^57q^{4} + z^58q^{5} + z^56q^{6} + q^{7} + z^85q^{8} + 6q^{9} + z^18q^{10} + z^33q^{12} + z^67q^{13} + z^80q^{14} + z^34q^{15} + z^81q^{16} + z^56q^{17} + z^68q^{18} + z^9q^{19} + z^115q^{20} + 3q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^68q^{2} + q^{3} + z^33q^{4} + z^10q^{5} + z^68q^{6} + 8q^{7} + z^49q^{8} + 8q^{9} + z^78q^{10} + z^110q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^10)\omega^{0}&0\\0&u(z^110)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(27083839005499675565*11 + O(11^20))*a - 271291718417836041021 + O(11^20)q^{3} + (162292480428689336571 + O(11^20))*a + 188072936790349862776 + O(11^20)q^{4} + (328002264841053264650 + O(11^20))*a - 169306127708261495964 + O(11^20)q^{5} + -(17522993953437238439 + O(11^20))*a - 29862709479915471587*11 + O(11^20)q^{6} + 282475249q^{7} + -(204817257092035286065 + O(11^20))*a + 75006667467933851464 + O(11^20)q^{8} + -(2600856497265273682*11 + O(11^20))*a - 328908677083270862528 + O(11^20)q^{9} + (326908494271874684189 + O(11^20))*a + 322829047537154051827 + O(11^20)q^{10} + (5193493114207165986*11 + O(11^20))*a + 16679766549346378529*11 + O(11^20)q^{11} + -(66280093307907788092 + O(11^20))*a - 29024485236007722877 + O(11^20)q^{12} + -(201658518870666665014 + O(11^20))*a - 37550823497101441392 + O(11^20)q^{13} + 282475249*aq^{14} + -(321917918696564702047 + O(11^20))*a - 298625959286463341745 + O(11^20)q^{15} + (266939708313810257645 + O(11^20))*a + 269800052534467704941 + O(11^20)q^{16} + (12714531000015928471 + O(11^20))*a + 4153171437614408898*11 + O(11^20)q^{17} + (198562101608725710433 + O(11^20))*a + 8797109329197787109*11 + O(11^20)q^{18} + (180626245645746674248 + O(11^20))*a + 279751405607502672069 + O(11^20)q^{19} + (11703271630058028030 + O(11^20))*a - 306518262898986637781 + O(11^20)q^{20} + -(1399551831717892865*11 + O(11^20))*a - 253450045493771948467 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -162292480428689336571 + O(11^20)x + -188072936790351959928 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^40q^{2} + 3q^{3} + z^27q^{4} + z^38q^{5} + z^16q^{6} + q^{7} + z^95q^{8} + 6q^{9} + z^78q^{10} + z^3q^{12} + z^17q^{13} + z^40q^{14} + z^14q^{15} + z^51q^{16} + z^16q^{17} + z^28q^{18} + z^99q^{19} + z^65q^{20} + 3q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^28q^{2} + q^{3} + z^3q^{4} + z^110q^{5} + z^28q^{6} + 8q^{7} + z^59q^{8} + 8q^{9} + z^18q^{10} + z^10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^110)\omega^{0}&0\\0&u(z^10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(27083839005499675565*11 + O(11^20))*a - 271291718417836041021 + O(11^20)q^{3} + (162292480428689336571 + O(11^20))*a + 188072936790349862776 + O(11^20)q^{4} + (328002264841053264650 + O(11^20))*a - 169306127708261495964 + O(11^20)q^{5} + -(17522993953437238439 + O(11^20))*a - 29862709479915471587*11 + O(11^20)q^{6} + 282475249q^{7} + -(204817257092035286065 + O(11^20))*a + 75006667467933851464 + O(11^20)q^{8} + -(2600856497265273682*11 + O(11^20))*a - 328908677083270862528 + O(11^20)q^{9} + (326908494271874684189 + O(11^20))*a + 322829047537154051827 + O(11^20)q^{10} + (5193493114207165986*11 + O(11^20))*a + 16679766549346378529*11 + O(11^20)q^{11} + -(66280093307907788092 + O(11^20))*a - 29024485236007722877 + O(11^20)q^{12} + -(201658518870666665014 + O(11^20))*a - 37550823497101441392 + O(11^20)q^{13} + 282475249*aq^{14} + -(321917918696564702047 + O(11^20))*a - 298625959286463341745 + O(11^20)q^{15} + (266939708313810257645 + O(11^20))*a + 269800052534467704941 + O(11^20)q^{16} + (12714531000015928471 + O(11^20))*a + 4153171437614408898*11 + O(11^20)q^{17} + (198562101608725710433 + O(11^20))*a + 8797109329197787109*11 + O(11^20)q^{18} + (180626245645746674248 + O(11^20))*a + 279751405607502672069 + O(11^20)q^{19} + (11703271630058028030 + O(11^20))*a - 306518262898986637781 + O(11^20)q^{20} + -(1399551831717892865*11 + O(11^20))*a - 253450045493771948467 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -162292480428689336571 + O(11^20)x + -188072936790351959928 + O(11^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 6q^{2} + q^{3} + 6q^{4} + 8q^{5} + 6q^{6} + 7q^{7} + 10q^{8} + 7q^{9} + 4q^{10} + 6q^{12} + 6q^{13} + 9q^{14} + 8q^{15} + q^{16} + 9q^{17} + 9q^{18} + 6q^{19} + 4q^{20} + 7q^{21} + \cdots \in S_{14}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 10q^{2} + 9q^{3} + 4q^{4} + 5q^{5} + 2q^{6} + 4q^{7} + 4q^{8} + 10q^{9} + 6q^{10} + 3q^{11} + 3q^{12} + 6q^{13} + 7q^{14} + q^{15} + 8q^{16} + 10q^{17} + q^{18} + 9q^{20} + 3q^{21} + \cdots \in S_{14}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 8q^{3} + 2q^{4} + 10q^{5} + 8q^{6} + 6q^{7} + 3q^{8} + 8q^{9} + 10q^{10} + q^{11} + 5q^{12} + 3q^{13} + 6q^{14} + 3q^{15} + 5q^{16} + 5q^{17} + 8q^{18} + q^{19} + 9q^{20} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^49q^{2} + z^41q^{3} + z^117q^{4} + z^46q^{5} + z^90q^{6} + 5q^{7} + q^{8} + z^79q^{9} + z^95q^{10} + z^30q^{11} + z^38q^{12} + z^34q^{13} + z^97q^{14} + z^87q^{15} + z^15q^{16} + z^106q^{17} + z^8q^{18} + z^69q^{19} + z^43q^{20} + z^89q^{21} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^59q^{2} + z^91q^{3} + z^87q^{4} + z^26q^{5} + z^30q^{6} + 5q^{7} + q^{8} + z^29q^{9} + z^85q^{10} + z^90q^{11} + z^58q^{12} + z^14q^{13} + z^107q^{14} + z^117q^{15} + z^45q^{16} + z^86q^{17} + z^88q^{18} + z^39q^{19} + z^113q^{20} + z^19q^{21} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 2q^{3} + 7q^{4} + 4q^{5} + 10q^{6} + 2q^{7} + 6q^{9} + 9q^{10} + 9q^{11} + 3q^{12} + 7q^{13} + 10q^{14} + 8q^{15} + 6q^{16} + 6q^{17} + 8q^{18} + 7q^{19} + 6q^{20} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^79q^{2} + z^31q^{3} + z^87q^{4} + z^90q^{5} + z^110q^{6} + 9q^{7} + 7q^{8} + z^19q^{9} + z^49q^{10} + z^110q^{11} + z^118q^{12} + z^63q^{13} + z^31q^{14} + zq^{15} + z^9q^{16} + z^13q^{17} + z^98q^{18} + 4q^{19} + z^57q^{20} + z^103q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^29q^{2} + z^101q^{3} + z^117q^{4} + z^30q^{5} + z^10q^{6} + 9q^{7} + 7q^{8} + z^89q^{9} + z^59q^{10} + z^10q^{11} + z^98q^{12} + z^93q^{13} + z^101q^{14} + z^11q^{15} + z^99q^{16} + z^23q^{17} + z^118q^{18} + 4q^{19} + z^27q^{20} + z^53q^{21} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^68q^{2} + q^{3} + z^33q^{4} + z^10q^{5} + z^68q^{6} + 8q^{7} + z^49q^{8} + 8q^{9} + z^78q^{10} + z^110q^{11} + z^33q^{12} + z^55q^{13} + z^104q^{14} + z^10q^{15} + z^33q^{16} + z^68q^{17} + z^104q^{18} + z^93q^{19} + z^43q^{20} + 8q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^28q^{2} + q^{3} + z^3q^{4} + z^110q^{5} + z^28q^{6} + 8q^{7} + z^59q^{8} + 8q^{9} + z^18q^{10} + z^10q^{11} + z^3q^{12} + z^5q^{13} + z^64q^{14} + z^110q^{15} + z^3q^{16} + z^28q^{17} + z^64q^{18} + z^63q^{19} + z^113q^{20} + 8q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^1104q^{2} + z^1030q^{3} + z^220q^{4} + z^1271q^{5} + z^804q^{6} + 3q^{7} + z^345q^{8} + z^1200q^{9} + z^1045q^{10} + z^1070q^{11} + z^1250q^{12} + z^461q^{13} + z^838q^{14} + z^971q^{15} + z^168q^{16} + z^618q^{17} + z^974q^{18} + z^507q^{19} + z^161q^{20} + z^764q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^174q^{2} + z^690q^{3} + z^1090q^{4} + z^681q^{5} + z^864q^{6} + 3q^{7} + z^1135q^{8} + z^1230q^{9} + z^855q^{10} + z^1130q^{11} + z^450q^{12} + z^1081q^{13} + z^1238q^{14} + z^41q^{15} + z^518q^{16} + z^148q^{17} + z^74q^{18} + z^257q^{19} + z^441q^{20} + z^424q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^584q^{2} + z^940q^{3} + z^20q^{4} + z^841q^{5} + z^194q^{6} + 3q^{7} + z^515q^{8} + z^230q^{9} + z^95q^{10} + z^460q^{11} + z^960q^{12} + z^1251q^{13} + z^318q^{14} + z^451q^{15} + z^378q^{16} + z^298q^{17} + z^814q^{18} + z^167q^{19} + z^861q^{20} + z^674q^{21} + \cdots \in S_{20}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{3} + 9q^{4} + 10q^{5} + 10q^{7} + 6q^{9} + 10q^{11} + 6q^{12} + 7q^{13} + 3q^{15} + 4q^{16} + 2q^{17} + 5q^{19} + 2q^{20} + 3q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 2q^{3} + 10q^{4} + 9q^{5} + 2q^{6} + 10q^{7} + 8q^{8} + q^{9} + 9q^{10} + q^{11} + 9q^{12} + 4q^{13} + 10q^{14} + 7q^{15} + 10q^{16} + 4q^{17} + q^{18} + 2q^{20} + 9q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 9q^{4} + 3q^{5} + q^{7} + 9q^{9} + 10q^{11} + 9q^{12} + 7q^{13} + 3q^{15} + 4q^{16} + 5q^{17} + 2q^{19} + 5q^{20} + q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 5q^{3} + 3q^{4} + 9q^{5} + 2q^{6} + q^{7} + 7q^{8} + 8q^{10} + 10q^{11} + 4q^{12} + 8q^{13} + 7q^{14} + q^{15} + 10q^{16} + 3q^{17} + 5q^{19} + 5q^{20} + 5q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 7q^{4} + 6q^{5} + q^{6} + q^{7} + 4q^{8} + 2q^{9} + 7q^{10} + 6q^{12} + 3q^{13} + 3q^{14} + 2q^{15} + 9q^{16} + 7q^{17} + 6q^{18} + 9q^{19} + 9q^{20} + 4q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 4q^{2} + 8q^{3} + 3q^{4} + 9q^{5} + 10q^{6} + q^{7} + 4q^{8} + 6q^{9} + 3q^{10} + 10q^{11} + 2q^{12} + 5q^{13} + 4q^{14} + 6q^{15} + 10q^{16} + 6q^{17} + 2q^{18} + 10q^{19} + 5q^{20} + 8q^{21} + \cdots \in S_{22}(\Gamma_0(7);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 8, \quad \ell = 11}$ \quad (19 forms)}

\begin{enumerate}
\item  Consider
$f = q + 10q^{3} + 4q^{5} + 5q^{7} + 7q^{9} + 4q^{13} + 7q^{15} + 10q^{17} + 3q^{19} + 6q^{21} + \cdots \in S_{14}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{3} + 5q^{5} + q^{7} + 10q^{9} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -12q^{3} + -4330q^{5} + -139992q^{7} + -1594179q^{9} + -589484*11q^{11} + -22588034q^{13} + 51960q^{15} + -23732270q^{17} + 325344836q^{19} + 1679904q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{3} + 9q^{5} + 2q^{7} + 8q^{15} + 6q^{17} + 3q^{21} + \cdots \in S_{14}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 7q^{3} + 9q^{5} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{3}$. The form
$f_0 = q + aq^{3} + 12*a + 4006q^{5} + -222*a + 152256q^{7} + 872*a + 1414557q^{9} + 531*a + 8005504q^{11} + -10452*a + 13929358q^{13} + 14470*a + 36106560q^{15} + 38184*a - 93545038q^{17} + -8529*11*a - 1678976*11q^{19} + -41328*a - 667971360q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -872x + -3008880$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 5q^{3} + 10q^{5} + 7q^{7} + 2q^{9} + 10q^{13} + 6q^{15} + 3q^{17} + 7q^{19} + 2q^{21} + \cdots \in S_{16}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 4q^{3} + 6q^{5} + 6q^{7} + 7q^{9} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2700q^{3} + -251890q^{5} + 1374072q^{7} + -7058907q^{9} + -3935156*11q^{11} + -323161466q^{13} + -680103000q^{15} + -191653646q^{17} + -6515456644q^{19} + 3709994400q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{3} + q^{5} + 3q^{7} + 10q^{15} + 3q^{17} + 8q^{21} + \cdots \in S_{16}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 7q^{3} + 9q^{5} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{15}$. The form
$f_0 = q + -3444q^{3} + 313358q^{5} + -2324616q^{7} + -226161*11q^{9} + -456604*11^2q^{11} + -10023598*11q^{13} + -1079204952q^{15} + -2601428750q^{17} + 177465844*11q^{19} + 8005977504q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 9q^{5} + 2q^{7} + 10q^{9} + 10q^{13} + 2q^{17} + 8q^{19} + \cdots \in S_{16}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + q^{5} + 8q^{7} + 2q^{9} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(4)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 29659132878511889008*11 + O(11^20)q^{3} + 308266706037187893917 + O(11^20)q^{5} + -238458227305870040588 + O(11^20)q^{7} + 189360097501490988236 + O(11^20)q^{9} + 2366594861980184581*11 + O(11^20)q^{11} + -145163955559332709737 + O(11^20)q^{13} + 29743906027638592201*11 + O(11^20)q^{15} + -96112956299898508947 + O(11^20)q^{17} + 3473901814486932800936 + O(11^21)q^{19} + 6385330492892579432*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{3} + 4q^{5} + 3q^{7} + 5q^{9} + q^{13} + 9q^{15} + 5q^{17} + 2q^{19} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 9q^{3} + 3q^{5} + 9q^{7} + 3q^{9} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 237853258544855010238 + O(11^20)q^{3} + 143445619107540422809 + O(11^20)q^{5} + -151011389797562871838 + O(11^20)q^{7} + 280446157570764627678 + O(11^20)q^{9} + -6128292804387041005*11 + O(11^20)q^{11} + 136748187992773101625 + O(11^20)q^{13} + -181024828818751938484 + O(11^20)q^{15} + -335098537127383232363 + O(11^20)q^{17} + 779378992030693016008 + O(11^21)q^{19} + 306402192853309311070 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{3} + 4q^{5} + 8q^{7} + 5q^{9} + 10q^{13} + 9q^{15} + 6q^{17} + 9q^{19} + 7q^{21} + \cdots \in S_{18}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 9q^{3} + 3q^{5} + 9q^{7} + 3q^{9} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{6}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -27120212785074008683 + O(11^20)q^{3} + -174075515412647438993 + O(11^20)q^{5} + -151823049696759149382 + O(11^20)q^{7} + -203258971072395800392 + O(11^20)q^{9} + -1912073832274955453*11 + O(11^20)q^{11} + 318101605297731572008 + O(11^20)q^{13} + 321580757079223955300 + O(11^20)q^{15} + 269735114044000647664 + O(11^20)q^{17} + 220005465942469156519 + O(11^20)q^{19} + -173162390089023029182 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{5} + q^{7} + 7q^{9} + 4q^{13} + 5q^{17} + 5q^{19} + \cdots \in S_{20}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + q^{5} + 8q^{7} + 2q^{9} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{8}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 28762896031311307819*11 + O(11^20)q^{3} + 879258209361647217272 + O(11^21)q^{5} + -2856029319604017868398 + O(11^21)q^{7} + 59689194141760551716 + O(11^20)q^{9} + -17682337531174718495*11 + O(11^20)q^{11} + -292846419252694870801 + O(11^20)q^{13} + 30133086800404770211*11 + O(11^20)q^{15} + 88133675594113824405 + O(11^20)q^{17} + -246354975700501317689 + O(11^20)q^{19} + 8799218661165635436*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{3} + 3q^{5} + 7q^{7} + 6q^{15} + 2q^{17} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 7q^{3} + 9q^{5} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{99}$. The form
$f_0 = q + aq^{3} + (71739913100518356081 + O(11^20))*a - 286771297198932320206 + O(11^20)q^{5} + -(89298272145247099096 + O(11^20))*a - 319324010414393401097 + O(11^20)q^{7} + -(3012953843206723610 + O(11^20))*a + 40305095844010507117 + O(11^20)q^{9} + -(261868908749327877655 + O(11^20))*a + 306917766934413219031 + O(11^20)q^{11} + -(146960528936824745075 + O(11^20))*a + 283648462680551198426 + O(11^20)q^{13} + (49429855242292176837 + O(11^21))*a + 2114529052840512092740 + O(11^21)q^{15} + -(5222660501278696115 + O(11^20))*a - 183128850258256009838 + O(11^20)q^{17} + (9252652515355046232*11 + O(11^20))*a + 526357107575210889*11^2 + O(11^20)q^{19} + (47007052107242204452 + O(11^20))*a + 61024949400868555464 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 3012953843206723610 + O(11^20)x + -40305095845172768584 + O(11^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + q^{3} + 8q^{5} + 9q^{7} + 8q^{9} + 4q^{13} + 8q^{15} + 2q^{17} + 3q^{19} + 9q^{21} + \cdots \in S_{20}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 4q^{3} + 6q^{5} + 6q^{7} + 7q^{9} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{8}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 3012953843206747342 + O(11^20)q^{3} + 237341441956642283587 + O(11^20)q^{5} + 272316958307207048365 + O(11^20)q^{7} + 151616048286617080251 + O(11^20)q^{9} + 13055498409929419156*11 + O(11^20)q^{11} + 293985161745674099474 + O(11^20)q^{13} + -6080281819353922040 + O(11^20)q^{15} + -106390654215567499567 + O(11^20)q^{17} + 133460784114714809888 + O(11^20)q^{19} + 241850928540593178288 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{3} + 4q^{5} + 5q^{7} + 2q^{15} + q^{17} + 8q^{21} + \cdots \in S_{22}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 7q^{3} + 9q^{5} + 2q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{111}$. The form
$f_0 = q + 192011347013165958526 + O(11^20)q^{3} + 196273029332040884436 + O(11^20)q^{5} + -8163195851865990623 + O(11^20)q^{7} + 22955383498403253067*11 + O(11^20)q^{9} + -650348885411941718*11^2 + O(11^20)q^{11} + -117066435239381991577*11 + O(11^21)q^{13} + 94228476522828356579 + O(11^20)q^{15} + -162433658798202972476 + O(11^20)q^{17} + 26058816584734320422*11 + O(11^20)q^{19} + -235635184637186998876 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 7q^{3} + 3q^{5} + 7q^{7} + 2q^{9} + 2q^{13} + 10q^{15} + 9q^{17} + 10q^{19} + 5q^{21} + \cdots \in S_{22}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{3} + 5q^{5} + q^{7} + 10q^{9} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{0}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -54482294323998791782 + O(11^20)q^{3} + 325033101411364126993 + O(11^20)q^{5} + -36661258535087029210 + O(11^20)q^{7} + 165475433795445166417 + O(11^20)q^{9} + 16179817711094748369*11 + O(11^20)q^{11} + -322845341485774539144 + O(11^20)q^{13} + -329062144376499511957 + O(11^20)q^{15} + 267123841190985321431 + O(11^20)q^{17} + 70862335466729695272 + O(11^20)q^{19} + 180401648425834546243 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 9q^{3} + 3q^{5} + 9q^{7} + 3q^{9} + 3q^{11} + 5q^{13} + 5q^{15} + q^{17} + 8q^{19} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{5} + 8q^{7} + 2q^{9} + 4q^{11} + 2q^{13} + 10q^{17} + 2q^{19} + \cdots \in S_{18}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{3} + 6q^{5} + 6q^{7} + 7q^{9} + 6q^{11} + 2q^{13} + 2q^{15} + 4q^{17} + 10q^{19} + 2q^{21} + \cdots \in S_{18}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{3} + 5q^{5} + q^{7} + 10q^{9} + q^{11} + q^{13} + 8q^{15} + 7q^{17} + 4q^{19} + 6q^{21} + \cdots \in S_{20}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{3} + 8q^{5} + 9q^{7} + 6q^{9} + 10q^{11} + 9q^{15} + 5q^{17} + 4q^{19} + 6q^{21} + \cdots \in S_{22}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^97q^{3} + z^116q^{5} + z^49q^{7} + z^76q^{9} + 10q^{11} + z^59q^{13} + z^93q^{15} + 2q^{17} + 7q^{19} + z^26q^{21} + \cdots \in S_{22}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^107q^{3} + z^76q^{5} + z^59q^{7} + z^116q^{9} + 10q^{11} + z^49q^{13} + z^63q^{15} + 2q^{17} + 7q^{19} + z^46q^{21} + \cdots \in S_{22}(\Gamma_0(8);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 9, \quad \ell = 11}$ \quad (28 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{2} + 4q^{4} + 4q^{5} + 7q^{7} + 7q^{8} + 4q^{10} + q^{13} + 7q^{14} + 8q^{16} + 3q^{17} + 6q^{19} + 5q^{20} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 3q^{4} + 5q^{5} + 8q^{7} + 2q^{8} + 4q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(9)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 12q^{2} + -8048q^{4} + 30210q^{5} + 235088q^{7} + -194880q^{8} + 362520q^{10} + 1016628*11q^{11} + 8049614q^{13} + 2821056q^{14} + 63590656q^{16} + 117494622q^{17} + -214061380q^{19} + -243130080q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{4} + 9q^{7} + 7q^{13} + 9q^{16} + q^{19} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{4} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{3}$. The form
$f_0 = q + aq^{2} + -272q^{4} + -520*aq^{5} + -133300q^{7} + -8464*aq^{8} + -374400*11q^{10} + 90400*aq^{11} + -30477550q^{13} + -133300*aq^{14} + -64806656q^{16} + -58992*11*aq^{17} + 81793064q^{19} + 141440*aq^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -720*11$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 5q^{2} + 6q^{4} + 3q^{5} + q^{7} + q^{8} + 4q^{10} + 6q^{13} + 5q^{14} + q^{16} + 2q^{17} + 6q^{19} + 7q^{20} + \cdots \in S_{14}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 4q^{2} + 10q^{4} + q^{5} + 9q^{7} + 5q^{8} + 4q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(10)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 54*a + 800*11q^{4} + 128*a - 23814q^{5} + 3456*a - 103816q^{7} + 3524*a + 917568q^{8} + -16902*a + 2174976q^{10} + -3328*11*a + 59292*11q^{11} + -89856*a + 11192414q^{13} + 7528*11*a + 58724352q^{14} + 665496*a - 12209792q^{16} + 72960*a - 43889202q^{17} + -767232*a + 148862036q^{19} + 213692*a - 92114496q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -54x + -16992$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{4} + 8q^{7} + 3q^{13} + q^{16} + 8q^{19} + \cdots \in S_{16}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{4} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{15}$. The form
$f_0 = q + -32768q^{4} + 1244900q^{7} + 397771850q^{13} + 1073741824q^{16} + 7700827736q^{19} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

\item  Consider
$f = q + 3q^{2} + 10q^{4} + 10q^{5} + 6q^{7} + 8q^{10} + 8q^{13} + 7q^{14} + 10q^{16} + q^{17} + 5q^{19} + q^{20} + \cdots \in S_{16}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{2} + 7q^{4} + 6q^{5} + 2q^{7} + 8q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(2)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 234q^{2} + 21988q^{4} + -280710q^{5} + -1373344q^{7} + -229320*11q^{8} + -65686140q^{10} + -3093732*11q^{11} + 384022262q^{13} + -321362496q^{14} + -1310772464q^{16} + -1259207586q^{17} + -2499071020q^{19} + -6172251480q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^66q^{2} + 3q^{4} + z^78q^{5} + 10q^{7} + z^90q^{8} + 4q^{10} + 9q^{13} + z^6q^{14} + z^18q^{17} + z^54q^{20} + \cdots \in S_{16}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^18q^{2} + q^{4} + z^6q^{5} + 7q^{7} + z^66q^{8} + 4q^{10} + z^78q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^42)\omega^{1}&0\\0&u(z^78)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 87112q^{4} + 464*aq^{5} + -2591260q^{7} + 54344*aq^{8} + 55624320q^{10} + 9920*11*aq^{11} + -77911990q^{13} + -2591260*aq^{14} + 332752064*11q^{16} + 217248*aq^{17} + -88778744*11q^{19} + 40419968*aq^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -119880$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^6q^{2} + 3q^{4} + z^18q^{5} + 10q^{7} + z^30q^{8} + 4q^{10} + 9q^{13} + z^66q^{14} + z^78q^{17} + z^114q^{20} + \cdots \in S_{16}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^78q^{2} + q^{4} + z^66q^{5} + 7q^{7} + z^6q^{8} + 4q^{10} + z^18q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^102)\omega^{1}&0\\0&u(z^18)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 87112q^{4} + 464*aq^{5} + -2591260q^{7} + 54344*aq^{8} + 55624320q^{10} + 9920*11*aq^{11} + -77911990q^{13} + -2591260*aq^{14} + 332752064*11q^{16} + 217248*aq^{17} + -88778744*11q^{19} + 40419968*aq^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -119880$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 5q^{4} + 3q^{5} + 5q^{7} + 2q^{8} + 8q^{10} + 6q^{14} + 7q^{16} + q^{17} + 4q^{19} + 4q^{20} + \cdots \in S_{18}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 6q^{2} + 4q^{4} + 5q^{5} + 4q^{7} + 8q^{8} + 8q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 295774179405406426504 + O(11^20)q^{2} + 1916136095384424997528 + O(11^21)q^{4} + 2310130633369951739801 + O(11^21)q^{5} + 96109659206687111045 + O(11^20)q^{7} + -48487063722199306851 + O(11^20)q^{8} + 30106657498292977335 + O(11^20)q^{10} + 24917664534297148523*11 + O(11^21)q^{11} + 96133497372782398653*11 + O(11^21)q^{13} + -3004827105011247347903 + O(11^21)q^{14} + -1852375435939028421997 + O(11^21)q^{16} + 233102293049293580716 + O(11^20)q^{17} + -298325460763110472501 + O(11^20)q^{19} + -1920776631590943762797 + O(11^21)q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 5q^{4} + 3q^{5} + 6q^{7} + 9q^{8} + 3q^{10} + 6q^{14} + 7q^{16} + 10q^{17} + 7q^{19} + 4q^{20} + \cdots \in S_{18}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 6q^{2} + 4q^{4} + 5q^{5} + 4q^{7} + 8q^{8} + 8q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{6}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -295774179405406427098 + O(11^20)q^{2} + -1916136095384424821012 + O(11^21)q^{4} + -2310130633369952122661 + O(11^21)q^{5} + -96109659206662639477 + O(11^20)q^{7} + 48487063722068966619 + O(11^20)q^{8} + -30106657499102359755 + O(11^20)q^{10} + -24917664534207370931*11 + O(11^21)q^{11} + -96133497373011434857*11 + O(11^21)q^{13} + 3004827105011682913199 + O(11^21)q^{14} + 1852375435989639745917 + O(11^21)q^{16} + -233102293014980454352 + O(11^20)q^{17} + 298325460843164014685 + O(11^20)q^{19} + 1920776632105469857877 + O(11^21)q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 6q^{4} + 8q^{5} + 3q^{7} + 3q^{10} + q^{13} + 8q^{14} + 8q^{16} + 8q^{17} + 10q^{19} + 4q^{20} + \cdots \in S_{20}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{2} + 7q^{4} + 6q^{5} + 2q^{7} + 8q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{8}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 191290808516859524879 + O(11^20)q^{2} + 263851407676615514982 + O(11^20)q^{4} + -93480575082868936053 + O(11^20)q^{5} + -225774540010305222868 + O(11^20)q^{7} + 7227616423302553478*11 + O(11^20)q^{8} + 230222509989842585614 + O(11^20)q^{10} + 4652872219213217838*11 + O(11^20)q^{11} + 201833677077681461484 + O(11^20)q^{13} + -318976276482900556072 + O(11^20)q^{14} + 33972979971878919738 + O(11^20)q^{16} + -134283332744169480116 + O(11^20)q^{17} + -204062495327584792427 + O(11^20)q^{19} + 2416328696856668277127 + O(11^21)q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{4} + 4q^{7} + 10q^{13} + 3q^{16} + 5q^{19} + \cdots \in S_{20}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{4} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{99}$. The form
$f_0 = q + aq^{2} + O(11^20)*a - 9127356037703374411 + O(11^20)q^{4} + -(50228065718999398322 + O(11^20))*a + O(11^21)q^{5} + O(11^20)*a + 119759928955378298679 + O(11^20)q^{7} + -(9127356037703898699 + O(11^20))*a + O(11^21)q^{8} + O(11^20)*a - 24712502156210551871*11 + O(11^20)q^{10} + -(167518910768740612545 + O(11^20))*a + O(11^21)q^{11} + O(11^20)*a - 280505939484285814916 + O(11^20)q^{13} + (119759928955378298679 + O(11^20))*a + O(11^21)q^{14} + O(11^20)*a - 136574039047327621814 + O(11^20)q^{16} + (28070630099113566267*11 + O(11^20))*a + O(11^21)q^{17} + O(11^20)*a + 87865861684397344905 + O(11^20)q^{19} + (247230513841791219412 + O(11^20))*a + O(11^21)q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + O(11^20)x + 829759639791168193*11 + O(11^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + z^30q^{2} + 4q^{4} + z^54q^{5} + 5q^{7} + z^102q^{8} + 7q^{10} + 8q^{13} + z^78q^{14} + z^54q^{17} + z^78q^{20} + \cdots \in S_{20}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^18q^{2} + q^{4} + z^6q^{5} + 7q^{7} + z^66q^{8} + 4q^{10} + z^78q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^42)\omega^{8}&0\\0&u(z^78)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + O(11^20)*a + 9127356037703772675 + O(11^20)q^{4} + (50228065718999424381 + O(11^20))*a + O(11^20)q^{5} + O(11^20)*a - 119759928955295162639 + O(11^20)q^{7} + (9127356037703248387 + O(11^20))*a + O(11^20)q^{8} + O(11^20)*a + 271837523716429532341 + O(11^20)q^{10} + (15228991888071561415*11 + O(11^20))*a + O(11^20)q^{11} + O(11^20)*a + 280505939510864653176 + O(11^20)q^{13} + -(119759928955295162639 + O(11^20))*a + O(11^20)q^{14} + O(11^20)*a + 12415821745236316322*11 + O(11^20)q^{16} + -(308776931088733071411 + O(11^20))*a + O(11^20)q^{17} + O(11^20)*a - 7987805436370603243*11 + O(11^20)q^{19} + -(247230513857340178644 + O(11^20))*a + O(11^20)q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + O(11^20)x + -9127356037704296963 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{4} + 6q^{7} + 9q^{13} + 4q^{16} + 7q^{19} + \cdots \in S_{22}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{4} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{111}$. The form
$f_0 = q + -2097152q^{4} + 1123983020q^{7} + -370076825230q^{13} + 4398046511104q^{16} + -35540635313176q^{19} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $Infinity$.

\item  Consider
$f = q + 6q^{2} + q^{4} + 3q^{5} + q^{7} + 5q^{8} + 7q^{10} + 6q^{13} + 6q^{14} + 6q^{16} + 6q^{17} + 9q^{19} + 3q^{20} + \cdots \in S_{22}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 3q^{4} + 5q^{5} + 8q^{7} + 2q^{8} + 4q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(9)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2844q^{2} + 5991184q^{4} + -3109950q^{5} + 363303920q^{7} + 11074627008q^{8} + -8844697800q^{10} + -1325621196*11q^{11} + 113350790702q^{13} + 1033236348480q^{14} + 18931815702784q^{16} + 8589389597982q^{17} + -29202939273796q^{19} + -18632282680800q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 7q^{4} + 5q^{5} + 8q^{7} + 7q^{8} + 7q^{10} + 3q^{13} + 9q^{14} + 9q^{16} + 4q^{17} + 9q^{19} + 2q^{20} + \cdots \in S_{22}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 4q^{2} + 10q^{4} + q^{5} + 9q^{7} + 5q^{8} + 4q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{0}&0\\0&u(10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -666*a + 3040*11^2q^{4} + 13568*a + 4019706q^{5} + 182016*a + 400559384q^{7} + -1285756*a - 1641684672q^{8} + -5016582*a + 33445011456q^{10} + 2842112*11*a - 9047627748*11q^{11} + 475236864*a + 134019420734q^{13} + 2308568*11^2*a + 448667983872q^{14} + 611332056*a - 3940794645632q^{16} + 1677304320*a + 6225306859278q^{17} + 22384332288*a + 13434275157716q^{19} + 8331896732*a - 20795768974656q^{20} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 666x + -2464992$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 6q^{2} + 4q^{4} + 5q^{5} + 4q^{7} + 8q^{8} + 8q^{10} + 3q^{11} + 2q^{14} + 8q^{16} + 9q^{17} + 5q^{19} + 9q^{20} + \cdots \in S_{16}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 6q^{4} + 8q^{5} + 10q^{7} + 6q^{8} + 10q^{10} + 7q^{13} + 7q^{14} + 8q^{16} + 7q^{17} + q^{19} + 4q^{20} + \cdots \in S_{16}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 5q^{2} + 7q^{4} + 6q^{5} + 2q^{7} + 8q^{10} + 2q^{11} + 6q^{13} + 10q^{14} + 6q^{16} + 5q^{17} + 4q^{19} + 9q^{20} + \cdots \in S_{18}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{4} + q^{5} + 5q^{16} + 4q^{20} + \cdots \in S_{18}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + z^78q^{2} + q^{4} + z^66q^{5} + 7q^{7} + z^6q^{8} + 4q^{10} + z^18q^{11} + 4q^{13} + z^42q^{14} + z^6q^{17} + z^66q^{20} + \cdots \in S_{18}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^18q^{2} + q^{4} + z^6q^{5} + 7q^{7} + z^66q^{8} + 4q^{10} + z^78q^{11} + 4q^{13} + z^102q^{14} + z^66q^{17} + z^6q^{20} + \cdots \in S_{18}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 8q^{4} + 2q^{5} + 5q^{7} + q^{8} + q^{10} + 5q^{13} + 8q^{14} + 2q^{16} + q^{17} + 2q^{19} + 5q^{20} + \cdots \in S_{20}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 4q^{2} + 10q^{4} + q^{5} + 9q^{7} + 5q^{8} + 4q^{10} + 10q^{11} + 7q^{13} + 3q^{14} + 4q^{16} + 8q^{17} + 8q^{19} + 10q^{20} + \cdots \in S_{20}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 3q^{4} + 5q^{5} + 8q^{7} + 2q^{8} + 4q^{10} + 9q^{11} + 3q^{13} + 2q^{14} + 10q^{16} + q^{17} + 8q^{19} + 4q^{20} + \cdots \in S_{20}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 2q^{4} + 10q^{5} + 9q^{7} + 9q^{10} + 10q^{11} + 4q^{13} + 7q^{14} + 7q^{16} + 2q^{17} + 9q^{20} + \cdots \in S_{22}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + 10q^{4} + 2q^{5} + 4q^{7} + 3q^{8} + 9q^{10} + 10q^{11} + 9q^{13} + 7q^{14} + 10q^{16} + 2q^{17} + 9q^{20} + \cdots \in S_{22}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + 10q^{4} + 7q^{5} + 9q^{7} + 3q^{8} + 4q^{10} + 10q^{11} + 9q^{13} + 2q^{14} + 10q^{16} + 2q^{17} + 5q^{19} + 4q^{20} + \cdots \in S_{22}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 10q^{4} + 4q^{5} + 9q^{7} + 8q^{8} + 4q^{10} + q^{11} + 9q^{13} + 9q^{14} + 10q^{16} + 9q^{17} + 5q^{19} + 7q^{20} + \cdots \in S_{22}(\Gamma_0(9);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 10, \quad \ell = 11}$ \quad (24 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + 7q^{3} + 4q^{4} + 6q^{5} + 3q^{6} + 8q^{8} + q^{10} + 6q^{12} + 2q^{13} + 9q^{15} + 5q^{16} + 2q^{17} + 3q^{19} + 2q^{20} + \cdots \in S_{14}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + q^{6} + 7q^{8} + q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(9)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -64q^{2} + -26q^{3} + 4096q^{4} + -15625q^{5} + 1664q^{6} + 48958*11q^{7} + -262144q^{8} + -144877*11q^{9} + 1000000q^{10} + -363168*11q^{11} + -106496q^{12} + -23834446q^{13} + -3133312*11q^{14} + 406250q^{15} + 16777216q^{16} + -192273222q^{17} + 9272128*11q^{18} + 166485740q^{19} + -64000000q^{20} + -1272908*11q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 3q^{6} + 3q^{7} + 3q^{8} + 10q^{10} + 5q^{12} + 5q^{13} + 5q^{14} + 2q^{15} + 5q^{16} + 9q^{17} + 2q^{19} + 2q^{20} + q^{21} + \cdots \in S_{14}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 9q^{3} + 3q^{4} + 2q^{5} + q^{6} + 5q^{7} + 4q^{8} + 10q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 64q^{2} + -2394q^{3} + 4096q^{4} + -15625q^{5} + -153216q^{6} + 438122q^{7} + 262144q^{8} + 376083*11q^{9} + -1000000q^{10} + -146208*11q^{11} + -9805824q^{12} + 2653106q^{13} + 28039808q^{14} + 37406250q^{15} + 16777216q^{16} + 108907962q^{17} + 24069312*11q^{18} + -63937300q^{19} + -64000000q^{20} + -1048864068q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 9q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + q^{10} + q^{12} + 5q^{13} + 9q^{14} + 5q^{15} + 3q^{16} + q^{18} + 3q^{19} + 4q^{20} + q^{21} + \cdots \in S_{16}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 6q^{6} + 7q^{7} + 8q^{8} + 4q^{9} + 10q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{4}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -128q^{2} + -5568q^{3} + 16384q^{4} + 78125q^{5} + 712704q^{6} + 2564996q^{7} + -2097152q^{8} + 16653717q^{9} + -10000000q^{10} + -7369788*11q^{11} + -91226112q^{12} + 351412022q^{13} + -328319488q^{14} + -435000000q^{15} + 268435456q^{16} + -196165914*11q^{17} + -2131675776q^{18} + -5107458100q^{19} + 1280000000q^{20} + -14281897728q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + 10q^{12} + 6q^{14} + 6q^{15} + 3q^{16} + 9q^{17} + 10q^{18} + 5q^{19} + 4q^{20} + 8q^{21} + \cdots \in S_{16}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + 128q^{2} + aq^{3} + 16384q^{4} + 78125q^{5} + 128*aq^{6} + 423*a - 102460q^{7} + 2097152q^{8} + -1844*a + 8757909q^{9} + 10000000q^{10} + -12006*a + 44706540q^{11} + 16384*aq^{12} + -28548*a + 118335542q^{13} + 54144*a - 13114880q^{14} + 78125*aq^{15} + 268435456q^{16} + -13428*a + 698489058q^{17} + -236032*a + 1121012352q^{18} + -134964*a + 2955266252q^{19} + 1280000000q^{20} + -882472*a + 9774183168q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 1844x + -23106816$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 3q^{2} + 6q^{3} + 9q^{4} + 4q^{5} + 7q^{6} + 6q^{7} + 5q^{8} + 5q^{9} + q^{10} + 10q^{12} + 7q^{14} + 2q^{15} + 4q^{16} + 10q^{17} + 4q^{18} + 7q^{19} + 3q^{20} + 3q^{21} + \cdots \in S_{18}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{17}$. The form
$f_0 = q + 256q^{2} + -14976q^{3} + 65536q^{4} + 390625q^{5} + -3833856q^{6} + 14808668q^{7} + 16777216q^{8} + 95140413q^{9} + 100000000q^{10} + -8967228*11^2q^{11} + -981467136q^{12} + -417754886*11q^{13} + 3791019008q^{14} + -5850000000q^{15} + 4294967296q^{16} + -16104698622q^{17} + 24355945728q^{18} + 48093117860q^{19} + 25600000000q^{20} + -221774611968q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 8q^{2} + 6q^{3} + 9q^{4} + 4q^{5} + 4q^{6} + 5q^{7} + 6q^{8} + 5q^{9} + 10q^{10} + 10q^{12} + 7q^{14} + 2q^{15} + 4q^{16} + q^{17} + 7q^{18} + 4q^{19} + 3q^{20} + 8q^{21} + \cdots \in S_{18}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{77}$. The form
$f_0 = q + -256q^{2} + aq^{3} + 65536q^{4} + 390625q^{5} + -256*aq^{6} + 2907*a + 2203100q^{7} + -16777216q^{8} + -1308*a - 54614979q^{9} + -100000000q^{10} + -5238*a - 239166300q^{11} + 65536*aq^{12} + -87228*a - 827964226q^{13} + -744192*a - 563993600q^{14} + 390625*aq^{15} + 4294967296q^{16} + 4460868*a + 18987357954q^{17} + 334848*a + 13981434624q^{18} + -7351668*a + 59528273828q^{19} + 25600000000q^{20} + -1599256*a + 216644709888q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 1308x + -74525184$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 8q^{2} + 10q^{3} + 9q^{4} + 7q^{5} + 3q^{6} + 6q^{8} + 3q^{9} + q^{10} + 2q^{12} + 4q^{13} + 4q^{15} + 4q^{16} + 9q^{17} + 2q^{18} + 8q^{20} + \cdots \in S_{18}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 7q^{3} + 5q^{4} + 8q^{5} + 5q^{6} + 2q^{8} + 4q^{9} + q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(2)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -256q^{2} + -188521619038109729967 + O(11^20)q^{3} + 65536q^{4} + -390625q^{5} + -176465161388229790920 + O(11^20)q^{6} + 14066004094758254895*11 + O(11^20)q^{7} + -16777216q^{8} + -227618148369827933011 + O(11^20)q^{9} + 100000000q^{10} + -335713506950885276839*11 + O(11^21)q^{11} + 100831654905305859053 + O(11^20)q^{12} + 114613112986167305954 + O(11^20)q^{13} + 7489288198344978049*11 + O(11^20)q^{14} + 24741458796981190312 + O(11^20)q^{15} + 4294967296q^{16} + 78774514299817932803 + O(11^20)q^{17} + -259003576456769949671 + O(11^20)q^{18} + -9292373287189110378*11 + O(11^20)q^{19} + -25600000000q^{20} + 2667967109634342535*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 7q^{3} + 9q^{4} + 7q^{5} + q^{6} + 10q^{7} + 6q^{8} + 7q^{9} + q^{10} + 8q^{12} + q^{13} + 3q^{14} + 5q^{15} + 4q^{16} + 7q^{17} + q^{18} + 3q^{19} + 8q^{20} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 6q^{3} + 5q^{4} + 8q^{5} + 2q^{6} + 3q^{7} + 9q^{8} + 2q^{9} + 10q^{10} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{6}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -256q^{2} + 188521619038109723659 + O(11^20)q^{3} + 65536q^{4} + -390625q^{5} + 176465161388231405768 + O(11^20)q^{6} + -154726045042334260001 + O(11^20)q^{7} + -16777216q^{8} + 227618148370051128917 + O(11^20)q^{9} + 100000000q^{10} + 335713506950993404903*11 + O(11^21)q^{11} + -100831654905719260141 + O(11^20)q^{12} + -114613112988185225182 + O(11^20)q^{13} + -82382170183469982603 + O(11^20)q^{14} + -24741458794517127812 + O(11^20)q^{15} + 4294967296q^{16} + -78774514318573572239 + O(11^20)q^{17} + 259003576399631797735 + O(11^20)q^{18} + 102216106022375383558 + O(11^20)q^{19} + -25600000000q^{20} + -29347638615729666261 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 10q^{3} + 9q^{4} + 7q^{5} + 8q^{6} + 5q^{8} + 3q^{9} + 10q^{10} + 2q^{12} + 7q^{13} + 4q^{15} + 4q^{16} + 2q^{17} + 9q^{18} + 8q^{20} + \cdots \in S_{18}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 7q^{3} + 5q^{4} + 8q^{5} + 5q^{6} + 2q^{8} + 4q^{9} + q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{6}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 256q^{2} + 123401660539921798077 + O(11^20)q^{3} + 65536q^{4} + -390625q^{5} + -28424663610340124735 + O(11^20)q^{6} + 28204035417009362968*11 + O(11^20)q^{7} + 16777216q^{8} + 323738380774909329364 + O(11^20)q^{9} + -100000000q^{10} + -328277058005406018879*11 + O(11^21)q^{11} + 123536060011088169051 + O(11^20)q^{12} + 32459729709241225145 + O(11^20)q^{13} + 3460393841480457470*11 + O(11^20)q^{14} + 108988500837405441927 + O(11^20)q^{15} + 4294967296q^{16} + -121218826518552859215 + O(11^20)q^{17} + 128776101671907185461 + O(11^20)q^{18} + -15008898930489994616*11 + O(11^20)q^{19} + -25600000000q^{20} + -14341529735651646387*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 7q^{3} + 9q^{4} + 7q^{5} + 10q^{6} + q^{7} + 5q^{8} + 7q^{9} + 10q^{10} + 8q^{12} + 10q^{13} + 3q^{14} + 5q^{15} + 4q^{16} + 4q^{17} + 10q^{18} + 8q^{19} + 8q^{20} + 7q^{21} + \cdots \in S_{18}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 6q^{3} + 5q^{4} + 8q^{5} + 2q^{6} + 3q^{7} + 9q^{8} + 2q^{9} + 10q^{10} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{1}&0\\0&u(5)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 256q^{2} + -123401660539921780449 + O(11^20)q^{3} + 65536q^{4} + -390625q^{5} + 28424663610344637503 + O(11^20)q^{6} + -310244389587075308452 + O(11^20)q^{7} + 16777216q^{8} + -323738380774673433298 + O(11^20)q^{9} + -100000000q^{10} + 328277058005411883903*11 + O(11^21)q^{11} + -123536060009932900443 + O(11^20)q^{12} + -32459729706345386677 + O(11^20)q^{13} + -38064332249197877994 + O(11^20)q^{14} + -108988500844291379427 + O(11^20)q^{15} + 4294967296q^{16} + 121218826520133071811 + O(11^20)q^{17} + -128776101611517792565 + O(11^20)q^{18} + 165097888279603653536 + O(11^20)q^{19} + -25600000000q^{20} + 157756827375816588201 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 7q^{3} + 3q^{4} + 9q^{5} + 2q^{6} + 2q^{7} + 4q^{8} + q^{9} + q^{10} + 10q^{12} + 10q^{14} + 8q^{15} + 9q^{16} + 6q^{17} + 5q^{18} + 10q^{19} + 5q^{20} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{89}$. The form
$f_0 = q + -512q^{2} + 38628q^{3} + 262144q^{4} + 1953125q^{5} + -19777536q^{6} + -144185776q^{7} + -134217728q^{8} + 329860917q^{9} + -1000000000q^{10} + -42615708*11^2q^{11} + 10126098432q^{12} + 312319618*11q^{13} + 73823117312q^{14} + 75445312500q^{15} + 68719476736q^{16} + 366347849874q^{17} + -168888789504q^{18} + -1604379002260q^{19} + 512000000000q^{20} + -5569608155328q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 6q^{2} + 4q^{3} + 3q^{4} + 9q^{5} + 2q^{6} + 8q^{7} + 7q^{8} + q^{9} + 10q^{10} + q^{12} + 2q^{13} + 4q^{14} + 3q^{15} + 9q^{16} + 6q^{18} + 6q^{19} + 5q^{20} + 10q^{21} + \cdots \in S_{20}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 6q^{6} + 7q^{7} + 8q^{8} + 4q^{9} + 10q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 512q^{2} + -62360943767820225117 + O(11^20)q^{3} + 262144q^{4} + 1953125q^{5} + -309553447293634827457 + O(11^20)q^{6} + -232736251188757922950 + O(11^20)q^{7} + 134217728q^{8} + -43983466786959330993 + O(11^20)q^{9} + 1000000000q^{10} + 25541891217659863966*11 + O(11^20)q^{11} + 277633789743130513452 + O(11^20)q^{12} + -181191391336971433350 + O(11^20)q^{13} + -84211505580934921823 + O(11^20)q^{14} + -22713963617755836379 + O(11^20)q^{15} + 68719476736q^{16} + 23681820091478238212*11 + O(11^20)q^{17} + -318785162148697164783 + O(11^20)q^{18} + -74596591293062429409 + O(11^20)q^{19} + 512000000000q^{20} + -163052511567518303094 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 6q^{3} + q^{4} + 10q^{5} + 6q^{6} + q^{8} + 10q^{10} + 6q^{12} + q^{13} + 5q^{15} + q^{16} + 4q^{17} + 10q^{19} + 10q^{20} + \cdots \in S_{22}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + q^{6} + 7q^{8} + q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(9)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1024q^{2} + 121796719666687454351 + O(11^20)q^{3} + 1048576q^{4} + -9765625q^{5} + 261091876164351553239 + O(11^20)q^{6} + -16408784589677705241*11 + O(11^20)q^{7} + 1073741824q^{8} + 1477520080877283846*11^2 + O(11^20)q^{9} + -10000000000q^{10} + 6188617856735597439*11 + O(11^20)q^{11} + 276333204069666863939 + O(11^20)q^{12} + 227145338352581703539 + O(11^20)q^{13} + 16154453484030063241*11 + O(11^20)q^{14} + 227795776492310874424 + O(11^20)q^{15} + 1099511627776q^{16} + -67864864861926321427 + O(11^20)q^{17} + 683053548451695472*11^2 + O(11^20)q^{18} + -1496707798305994099398 + O(11^21)q^{19} + -10240000000000q^{20} + 28737293916594244754*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 5q^{3} + q^{4} + 10q^{5} + 6q^{6} + 2q^{7} + 10q^{8} + q^{10} + 5q^{12} + 8q^{13} + 9q^{14} + 6q^{15} + q^{16} + 7q^{17} + 3q^{19} + 10q^{20} + 10q^{21} + \cdots \in S_{22}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 9q^{3} + 3q^{4} + 2q^{5} + q^{6} + 5q^{7} + 4q^{8} + 10q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{0}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1024q^{2} + -67199857022868093464 + O(11^20)q^{3} + 1048576q^{4} + -9765625q^{5} + 192154108295806768634 + O(11^20)q^{6} + 78892459375844053183 + O(11^20)q^{7} + -1073741824q^{8} + 273554294673459834*11^2 + O(11^20)q^{9} + 10000000000q^{10} + -20453552651828856666*11 + O(11^20)q^{11} + -322808374598608394524 + O(11^20)q^{12} + -62805754801694191730 + O(11^20)q^{13} + -55879008957109355272 + O(11^20)q^{14} + -179317907952060942872 + O(11^20)q^{15} + 1099511627776q^{16} + 141506906831414809956 + O(11^20)q^{17} + -2123732071011295966*11^2 + O(11^20)q^{18} + 87716077063810466142 + O(11^20)q^{19} + -10240000000000q^{20} + -86525995233260843327 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 6q^{6} + 7q^{7} + 8q^{8} + 4q^{9} + 10q^{10} + q^{11} + q^{12} + 8q^{13} + 3q^{14} + 4q^{15} + 5q^{16} + 8q^{18} + 10q^{19} + 9q^{20} + 10q^{21} + \cdots \in S_{14}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 6q^{3} + 5q^{4} + 8q^{5} + 2q^{6} + 3q^{7} + 9q^{8} + 2q^{9} + 10q^{10} + 5q^{11} + 8q^{12} + 6q^{13} + q^{14} + 4q^{15} + 3q^{16} + 3q^{17} + 8q^{18} + 10q^{19} + 7q^{20} + 7q^{21} + \cdots \in S_{16}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 7q^{3} + 5q^{4} + 8q^{5} + 5q^{6} + 2q^{8} + 4q^{9} + q^{10} + 2q^{11} + 2q^{12} + 9q^{13} + q^{15} + 3q^{16} + 4q^{17} + 6q^{18} + 7q^{20} + \cdots \in S_{16}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + q^{6} + 7q^{8} + q^{10} + 9q^{11} + 6q^{12} + 6q^{13} + 4q^{15} + 9q^{16} + 8q^{17} + 4q^{19} + 6q^{20} + \cdots \in S_{20}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 9q^{3} + 3q^{4} + 2q^{5} + q^{6} + 5q^{7} + 4q^{8} + 10q^{10} + 7q^{11} + 5q^{12} + 4q^{13} + 3q^{14} + 7q^{15} + 9q^{16} + 3q^{17} + 10q^{19} + 6q^{20} + q^{21} + \cdots \in S_{20}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 5q^{3} + 3q^{4} + 9q^{5} + 8q^{6} + 9q^{7} + 7q^{8} + 10q^{9} + 10q^{10} + q^{11} + 4q^{12} + 7q^{13} + 10q^{14} + q^{15} + 9q^{16} + 3q^{17} + 5q^{18} + 8q^{19} + 5q^{20} + q^{21} + \cdots \in S_{20}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 10q^{3} + q^{4} + q^{5} + 10q^{6} + 3q^{7} + q^{8} + 9q^{9} + q^{10} + q^{11} + 10q^{12} + 5q^{13} + 3q^{14} + 10q^{15} + q^{16} + 4q^{17} + 9q^{18} + 5q^{19} + q^{20} + 8q^{21} + \cdots \in S_{22}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{3} + q^{4} + 10q^{5} + q^{6} + 10q^{7} + q^{8} + 9q^{9} + 10q^{10} + 10q^{11} + q^{12} + 2q^{13} + 10q^{14} + 10q^{15} + q^{16} + 8q^{17} + 9q^{18} + 10q^{19} + 10q^{20} + 10q^{21} + \cdots \in S_{22}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + 5q^{3} + q^{4} + q^{5} + 6q^{6} + 6q^{7} + 10q^{8} + 10q^{10} + 10q^{11} + 5q^{12} + 2q^{13} + 5q^{14} + 5q^{15} + q^{16} + 4q^{17} + 9q^{19} + q^{20} + 8q^{21} + \cdots \in S_{22}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + q^{3} + q^{4} + 10q^{5} + 10q^{6} + 5q^{7} + 10q^{8} + 9q^{9} + q^{10} + q^{11} + q^{12} + 2q^{13} + 6q^{14} + 10q^{15} + q^{16} + 3q^{17} + 2q^{18} + 4q^{19} + 10q^{20} + 5q^{21} + \cdots \in S_{22}(\Gamma_0(10);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 12, \quad \ell = 11}$ \quad (14 forms)}

\begin{enumerate}
\item  Consider
$f = q + 8q^{3} + 2q^{7} + 9q^{9} + 4q^{19} + 5q^{21} + \cdots \in S_{14}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{3} + 7q^{7} + 5q^{9} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(9)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -729q^{3} + -1350*11q^{5} + -62896q^{7} + 531441q^{9} + 464076*11q^{11} + 94910*11^2q^{13} + 984150*11q^{15} + 10905894*11q^{17} + 332601020q^{19} + 45851184q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{3} + 8q^{5} + q^{7} + 4q^{9} + q^{13} + 5q^{15} + 10q^{17} + 8q^{19} + 2q^{21} + \cdots \in S_{16}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 6q^{3} + 7q^{5} + 4q^{7} + 3q^{9} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(2)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2187q^{3} + 45702q^{5} + 1217888q^{7} + 4782969q^{9} + -2445084*11q^{11} + -162581770q^{13} + -99950274q^{15} + -743272542q^{17} + -4003014700q^{19} + -2663521056q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{3} + 10q^{5} + 3q^{7} + 4q^{9} + 8q^{13} + 2q^{15} + 10q^{17} + 6q^{19} + 5q^{21} + \cdots \in S_{16}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{3} + 6q^{5} + q^{7} + 3q^{9} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2187q^{3} + 99368425518821242871 + O(11^20)q^{5} + -221565834736829617416 + O(11^20)q^{7} + 4782969q^{9} + -107420916240889308462*11 + O(11^21)q^{11} + -56612114814419702090 + O(11^20)q^{13} + 20498246445175186954 + O(11^20)q^{15} + 33375331517442434101 + O(11^20)q^{17} + -35379829084844793433 + O(11^20)q^{19} + -184484218003166664072 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{3} + 9q^{5} + q^{7} + 4q^{9} + 2q^{13} + 4q^{15} + 9q^{17} + 3q^{19} + 9q^{21} + \cdots \in S_{16}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 4q^{5} + 8q^{7} + 9q^{9} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{4}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2187q^{3} + -99368425518821173211 + O(11^20)q^{5} + 221565834736832108920 + O(11^20)q^{7} + 4782969q^{9} + 107420916240887947014*11 + O(11^21)q^{11} + 56612114814431459670 + O(11^20)q^{13} + -20498246445022840534 + O(11^20)q^{15} + -33375331513186356817 + O(11^20)q^{17} + 35379829094086482833 + O(11^20)q^{19} + 184484218008615583320 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{3} + 2q^{5} + 6q^{7} + 5q^{9} + 7q^{13} + 3q^{15} + 3q^{17} + 5q^{19} + 9q^{21} + \cdots \in S_{20}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 6q^{3} + 7q^{5} + 4q^{7} + 3q^{9} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{8}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -19683q^{3} + 295581888853621095563 + O(11^20)q^{5} + -90943751437688053778 + O(11^20)q^{7} + 387420489q^{9} + -7682885593580008699*11 + O(11^20)q^{11} + -225220569095124000237 + O(11^20)q^{13} + 3637870954935603719 + O(11^20)q^{15} + -281462457965651956925 + O(11^20)q^{17} + 195456618287351333896 + O(11^20)q^{19} + -141876967528221971487 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{3} + 8q^{5} + 7q^{7} + 5q^{9} + q^{13} + 10q^{15} + 3q^{17} + q^{19} + 6q^{21} + \cdots \in S_{20}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{3} + 6q^{5} + q^{7} + 3q^{9} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{8}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 19683q^{3} + -189034548010531130878 + O(11^20)q^{5} + -144518240427715935048 + O(11^20)q^{7} + 387420489q^{9} + -222581537913649827424*11 + O(11^21)q^{11} + -79660889634454646484 + O(11^20)q^{13} + 213213480705161819057 + O(11^20)q^{15} + 11395749718213555159 + O(11^20)q^{17} + -139993093474241427892 + O(11^20)q^{19} + -165547763869030647956 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{3} + 5q^{5} + 6q^{7} + 5q^{9} + 3q^{13} + 9q^{15} + 6q^{17} + 6q^{19} + 2q^{21} + \cdots \in S_{20}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{3} + 4q^{5} + 8q^{7} + 9q^{9} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 19683q^{3} + 189034548010527863770 + O(11^20)q^{5} + 144518240427688911064 + O(11^20)q^{7} + 387420489q^{9} + 222581537914620915256*11 + O(11^21)q^{11} + 79660889668122616384 + O(11^20)q^{13} + -213213480769468305821 + O(11^20)q^{15} + -11395749980939700403 + O(11^20)q^{17} + 139993093909559256812 + O(11^20)q^{19} + 165547763337117570884 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{3} + 5q^{7} + q^{9} + 6q^{19} + 6q^{21} + \cdots \in S_{22}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{3} + 7q^{7} + 5q^{9} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{0}&0\\0&u(9)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -59049q^{3} + -1188113947537070116*11^2 + O(11^20)q^{5} + -300638051359036864468 + O(11^20)q^{7} + 3486784401q^{9} + -4711345065774705928*11 + O(11^20)q^{11} + 135293671852073098*11^2 + O(11^20)q^{13} + 1903826471476452426*11^2 + O(11^20)q^{15} + -1175244497119713399*11^2 + O(11^20)q^{17} + 116266404642431588474 + O(11^20)q^{19} + -150571580625712825056 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 3q^{3} + 4q^{5} + 8q^{7} + 9q^{9} + 3q^{11} + q^{13} + q^{15} + 7q^{17} + 10q^{19} + 2q^{21} + \cdots \in S_{14}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{3} + 6q^{5} + q^{7} + 3q^{9} + 7q^{11} + 6q^{13} + 8q^{15} + 6q^{17} + 7q^{19} + 5q^{21} + \cdots \in S_{18}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{3} + 7q^{5} + 4q^{7} + 3q^{9} + 2q^{11} + 9q^{13} + 9q^{15} + 6q^{17} + 2q^{19} + 2q^{21} + \cdots \in S_{18}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{3} + 7q^{7} + 5q^{9} + 9q^{11} + 9q^{19} + 5q^{21} + \cdots \in S_{20}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{3} + 2q^{5} + 9q^{7} + q^{9} + q^{11} + 9q^{13} + 2q^{15} + 4q^{17} + 5q^{19} + 9q^{21} + \cdots \in S_{22}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{3} + 2q^{5} + 2q^{7} + q^{9} + 10q^{11} + 6q^{13} + 9q^{15} + 7q^{17} + 9q^{19} + 9q^{21} + \cdots \in S_{22}(\Gamma_0(12);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 13, \quad \ell = 11}$ \quad (83 forms)}

\begin{enumerate}
\item  Consider
$f = q + z^345q^{2} + z^873q^{3} + z^885q^{4} + z^413q^{5} + z^1218q^{6} + z^1009q^{7} + z^900q^{8} + z^533q^{9} + z^758q^{10} + z^428q^{12} + 2q^{13} + z^24q^{14} + z^1286q^{15} + z^268q^{16} + z^421q^{17} + z^878q^{18} + z^777q^{19} + z^1298q^{20} + z^552q^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^79q^{2} + z^75q^{3} + z^353q^{4} + z^679q^{5} + z^154q^{6} + z^477q^{7} + z^102q^{8} + z^267q^{9} + z^758q^{10} + z^304q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1026)\omega^{1}&0\\0&u(z^304)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (16005551973115477565 + O(11^19))*a^2 + (8243041758427855253 + O(11^19))*a + 14465006709606521*11^2 + O(11^19)q^{3} + a^2 - 8192q^{4} + -(9291558526035859107 + O(11^19))*a^2 - (18103262747053753326 + O(11^19))*a + 12047729521719131160 + O(11^19)q^{5} + -(16112201963386724258 + O(11^19))*a^2 - (29143970486797715143 + O(11^19))*a + 22730914427755667242 + O(11^19)q^{6} + (25443959180343833208 + O(11^19))*a^2 - (18930615978107892288 + O(11^19))*a + 16123129440440084632 + O(11^19)q^{7} + (683299281562938609 + O(11^20))*a^2 + (107502242810120141802 + O(11^20))*a + 201966393462920176474 + O(11^20)q^{8} + (21659806673762192790 + O(11^19))*a^2 + (23771163889408222101 + O(11^19))*a - 18647078376837362026 + O(11^19)q^{9} + (21455651864168806428 + O(11^19))*a^2 - (28335397182186992009 + O(11^19))*a - 10605109664198985297 + O(11^19)q^{10} + (2175340566621712294*11 + O(11^19))*a^2 + (201722484649890868*11^2 + O(11^19))*a + 1420064505915144472*11 + O(11^19)q^{11} + (2816608497017965885 + O(11^19))*a^2 - (17112447624288523339 + O(11^19))*a + 27859442041690006778 + O(11^19)q^{12} + -4826809q^{13} + (24220199749171452769 + O(11^19))*a^2 - (25785290802068765213 + O(11^19))*a + 28734640566894471713 + O(11^19)q^{14} + (1655834563361292579 + O(11^19))*a^2 - (12436641021993104489 + O(11^19))*a - 30161350824026259327 + O(11^19)q^{15} + (214186507981966739522 + O(11^20))*a^2 - (323849865490893615113 + O(11^20))*a + 131999550178696667116 + O(11^20)q^{16} + (20274037688345734416 + O(11^19))*a^2 - (2374970993421464829*11 + O(11^19))*a + 16707157294076172168 + O(11^19)q^{17} + -(5218876177048932641 + O(11^19))*a^2 + (1879662416312519028*11 + O(11^19))*a + 5012582914944999991 + O(11^19)q^{18} + -(1255269117748431068*11 + O(11^20))*a^2 + (230995534170507427749 + O(11^20))*a + 142221369404082563368 + O(11^20)q^{19} + -(4449672850706400853 + O(11^19))*a^2 - (15501463390119753492 + O(11^19))*a - 29766122654957670508 + O(11^19)q^{20} + (18612607575652097868 + O(11^19))*a^2 - (7683860872507135854 + O(11^19))*a - 16066917021893636873 + O(11^19)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -683299281562938609 + O(11^20)x^{2} + -107502242810120158186 + O(11^20)x + -201966393462920176474 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 7q^{3} + q^{4} + 4q^{5} + 10q^{6} + q^{8} + q^{10} + 7q^{12} + 2q^{13} + 6q^{15} + 6q^{16} + 2q^{17} + 10q^{19} + 4q^{20} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 9q^{2} + 2q^{3} + 9q^{4} + 5q^{5} + 7q^{6} + 5q^{8} + q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 304462516611205180781 + O(11^20)q^{2} + 23695381738779418664 + O(11^19)q^{3} + -256162498512043596802 + O(11^20)q^{4} + 3406677579710665814 + O(11^19)q^{5} + -1054325337858528737 + O(11^19)q^{6} + -1265540813087045007*11 + O(11^19)q^{7} + 33042100503307375899 + O(11^20)q^{8} + 87385427198050635*11^2 + O(11^19)q^{9} + 28065407621175705232 + O(11^19)q^{10} + 1462014581555199542*11 + O(11^19)q^{11} + 7257639367369664412 + O(11^19)q^{12} + -4826809q^{13} + 745061798067827230*11 + O(11^19)q^{14} + -12562532401668894518 + O(11^19)q^{15} + 184477290541173625584 + O(11^20)q^{16} + -25517247239265291673 + O(11^19)q^{17} + 179126587456794036*11^2 + O(11^19)q^{18} + -156584102871477226089 + O(11^20)q^{19} + 20486026408557701138 + O(11^19)q^{20} + -720670146443391721*11 + O(11^19)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4*z^9 + 10*z^8 + 7*z^7 + 7*z^6 + 6*z^5 + 10*z^4 + 8*z^3 + 8*z^2 + 10*z + 7q^{2} + 4*z^9 + 4*z^8 + 9*z^7 + 9*z^6 + z^4 + 2*z^3 + 5*z^2 + 2*z + 10q^{3} + 10*z^8 + 9*z^7 + z^6 + 3*z^5 + z^4 + 7*z^3 + 7*z^2 + zq^{4} + 10*z^9 + 9*z^8 + 7*z^7 + 4*z^5 + 6*z^4 + 3*z^3 + 10*z^2 + 5*z + 10q^{5} + z^9 + 10*z^8 + 9*z^7 + 10*z^6 + 9*z^5 + 2*z^4 + 4*z^3 + 7*z^2 + 8*z + 7q^{6} + 8*z^9 + 5*z^8 + 3*z^7 + 7*z^6 + 8*z^5 + z^3 + 8*z^2 + 4*z + 10q^{7} + 2*z^9 + 9*z^8 + 4*z^7 + z^6 + 10*z^5 + z^4 + 3*z^3 + 3*z^2 + 5*z + 9q^{8} + 10*z^9 + 4*z^8 + 6*z^7 + 2*z^6 + 4*z^5 + 7*z^4 + 8*z^3 + 7*z^2 + 6*z + 6q^{9} + 9*z^9 + 8*z^8 + 10*z^7 + 10*z^6 + 6*z^5 + 7*z^4 + z^3 + 7*z^2 + 2*zq^{10} + 8*z^9 + 5*z^8 + 2*z^7 + 9*z^6 + 8*z^5 + 6*z^4 + z^3 + 9*z^2 + 7*z + 3q^{12} + 9q^{13} + 5*z^9 + 2*z^8 + 5*z^6 + 4*z^5 + 2*z^4 + 8*z^3 + 10*z^2 + 6*zq^{14} + 6*z^9 + 10*z^8 + 4*z^7 + 7*z^6 + 9*z^5 + 5*z^4 + 6*z^3 + 9*z^2 + 5*z + 2q^{15} + 3*z^8 + 8*z^7 + 8*z^6 + 9*z^5 + 2*z^4 + 4*z^3 + 3*z^2 + 3*z + 4q^{16} + 8*z^9 + 6*z^8 + 8*z^7 + 8*z^6 + 8*z^5 + 4*z^4 + 6*z^2 + 7*z + 2q^{17} + 2*z^9 + 10*z^8 + 5*z^7 + 2*z^6 + 7*z^5 + 6*z^4 + 7*z^3 + 8*z^2 + 7*z + 7q^{18} + 7*z^9 + 5*z^8 + z^7 + 3*z^6 + z^5 + 7*z^4 + 2*z^3 + 6*z + 3q^{19} + z^9 + 10*z^8 + 8*z^7 + z^6 + 9*z^5 + 8*z^4 + 4*z^3 + 8*z^2q^{20} + 5*z^9 + 4*z^8 + 5*z^7 + 6*z^6 + 4*z^5 + 7*z^4 + 6*z^3 + 5*z + 5q^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^142133q^{2} + z^40440q^{3} + z^141494q^{4} + z^103104q^{5} + z^21523q^{6} + z^153127q^{7} + z^95697q^{8} + z^99297q^{9} + z^84187q^{10} + z^66312q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^94738)\omega^{1}&0\\0&u(z^66312)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(21387163858956230833 + O(11^20))*a^4 - (59404096813310740182 + O(11^20))*a^3 + (285434393449487570636 + O(11^20))*a^2 + (258382076893934702347 + O(11^20))*a + 325427090450479554154 + O(11^20)q^{3} + a^2 - 8192q^{4} + (149048911017397375406 + O(11^20))*a^4 - (212751438041340910991 + O(11^20))*a^3 + (169062434483015614325 + O(11^20))*a^2 - (93869867883927977009 + O(11^20))*a + 252179276043442089627 + O(11^20)q^{5} + (245758664138838808658 + O(11^20))*a^4 + (127053753102524643367 + O(11^20))*a^3 + (121601149095455710971 + O(11^20))*a^2 + (138425283053563602671 + O(11^20))*a + 212168488396872087078 + O(11^20)q^{6} + (216634659326732592466 + O(11^20))*a^4 + (17173076027588612481*11 + O(11^20))*a^3 - (79383888001421231121 + O(11^20))*a^2 - (168778358570009154338 + O(11^20))*a - 274123072264624336910 + O(11^20)q^{7} + a^3 - 16384*aq^{8} + -(292340748728231565075 + O(11^20))*a^4 - (311822893053470544917 + O(11^20))*a^3 - (157885957550417846730 + O(11^20))*a^2 + (228415951778337570761 + O(11^20))*a - 49953877595870578364 + O(11^20)q^{9} + -(19312600920288775336 + O(11^20))*a^4 - (131022188173597099255 + O(11^20))*a^3 + (273898172937446128592 + O(11^20))*a^2 - (58808135793201766318 + O(11^20))*a + 47512973853813980874 + O(11^20)q^{10} + (12633064790096194464*11 + O(11^20))*a^4 - (25533115118981745577*11 + O(11^20))*a^3 + (28779378768876882704*11 + O(11^20))*a^2 - (20699063029377833440*11 + O(11^20))*a - 24144466969759816095*11 + O(11^20)q^{11} + -(63849125614334263838 + O(11^20))*a^4 + (226173802006618836353 + O(11^20))*a^3 - (38968999222507754879 + O(11^20))*a^2 - (333800080229330717099 + O(11^20))*a - 1967432049382362515 + O(11^20)q^{12} + 4826809q^{13} + -(293667292172346286834 + O(11^20))*a^4 + (25674949162002874488 + O(11^20))*a^3 - (48914422817466958908 + O(11^20))*a^2 - (187333631007220538067 + O(11^20))*a + 164801161638778764219 + O(11^20)q^{14} + (180266126201550111797 + O(11^20))*a^4 - (14235042070596246561 + O(11^20))*a^3 - (273381884940763051701 + O(11^20))*a^2 + (175548833054252498936 + O(11^20))*a - 117119760516970272401 + O(11^20)q^{15} + a^4 - 24576*a^2 + 67108864q^{16} + -(263085504903913715282 + O(11^20))*a^4 + (86811378675964927121 + O(11^20))*a^3 + (59772151561100280682 + O(11^20))*a^2 - (278490724226963341212 + O(11^20))*a + 89678309788211630767 + O(11^20)q^{17} + (292960356135929954295 + O(11^20))*a^4 + (126788244543683647480 + O(11^20))*a^3 + (269630641206179273049 + O(11^20))*a^2 + (63933904165625109292 + O(11^20))*a - 40656570550301511864 + O(11^20)q^{18} + -(129342595797067967202 + O(11^20))*a^4 + (92833397036767088011 + O(11^20))*a^3 - (153708565622632542186 + O(11^20))*a^2 - (131759468696263651596 + O(11^20))*a + 208519509260609272666 + O(11^20)q^{19} + (155132964924173791298 + O(11^20))*a^4 + (164927169563368200405 + O(11^20))*a^3 - (207290028010266993472 + O(11^20))*a^2 + (88270279220978549808 + O(11^20))*a + 19657352748313031426 + O(11^20)q^{20} + -(95906054232732768975 + O(11^20))*a^4 + (66408531782613740267 + O(11^20))*a^3 + (143191402289721066832 + O(11^20))*a^2 + (141901363757969604243 + O(11^20))*a - 102563181908536027402 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -18405356653286492917 + O(11^20)x^{4} + -45435746779326533209 + O(11^20)x^{3} + 197200980531799123243 + O(11^20)x^{2} + -64509515765869694494 + O(11^20)x + -76188913029531034323 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10*z^8 + 5*z^7 + 5*z^6 + 7*z^5 + 8*z^4 + 4*z^3 + 7*z^2 + z + 6q^{2} + 8*z^8 + 4*z^7 + 6*z^6 + 8*z^5 + 5*z^4 + 8*z^3 + z^2 + 9*z + 6q^{3} + 9*z^9 + 9*z^8 + 10*z^7 + 4*z^6 + z^5 + 10*z^4 + z^3 + 4*z^2 + 2*z + 1q^{4} + 7*z^9 + 4*z^8 + 3*z^6 + 3*z^5 + 9*z^4 + 7*z^3 + 7*z^2 + 7*z + 1q^{5} + 5*z^9 + 2*z^8 + 4*z^7 + z^5 + 10*z^4 + 2*z^3 + 9*z^2 + 3*z + 9q^{6} + 9*z^9 + 3*z^8 + 8*z^7 + 7*z^6 + 2*z^5 + 9*z^4 + 9*z^3 + 9*z^2 + 7*zq^{7} + 8*z^9 + 8*z^8 + 2*z^7 + 7*z^6 + 10*z^5 + 6*z^4 + 2*z^3 + 5*z^2 + 3q^{8} + 9*z^9 + 8*z^8 + 8*z^7 + 10*z^6 + 10*z^5 + 2*z^4 + 9*z^3 + 5*z^2 + 7*z + 3q^{9} + 6*z^9 + 3*z^8 + 5*z^7 + 8*z^6 + 6*z^5 + 2*z^4 + 7*z^3 + z^2 + 6*z + 1q^{10} + 2*z^9 + 5*z^8 + 7*z^7 + 7*z^6 + 9*z^5 + 8*z^4 + 5*z^3 + 4*z + 5q^{12} + 9q^{13} + 9*z^9 + 5*z^8 + 10*z^7 + 6*z^6 + 10*z^5 + 9*z^4 + z^3 + 5*z^2 + 2*z + 2q^{14} + 2*z^9 + 8*z^8 + 10*z^7 + 9*z^6 + z^5 + 5*z^4 + 8*z^3 + 3*z^2 + 7*z + 1q^{15} + 2*z^9 + z^8 + 9*z^7 + 8*z^6 + 4*z^5 + 5*z^4 + 10*z^3 + 2*z^2 + 7*z + 9q^{16} + 7*z^9 + 8*z^8 + 9*z^7 + 10*z^6 + 2*z^4 + 8*z^3 + 2*z^2 + 6*z + 8q^{17} + 3*z^9 + z^8 + 2*z^7 + 3*z^6 + 5*z^5 + 6*z^4 + 8*z^3 + 3*z^2 + 6*z + 1q^{18} + 2*z^7 + 5*z^6 + 2*z^5 + 7*z^4 + 4*z^3 + 7*z^2 + 10*z + 2q^{19} + 9*z^9 + 4*z^8 + 3*z^7 + 8*z^5 + 9*z^4 + 9*z^3 + 3q^{20} + 3*z^9 + 3*z^8 + 7*z^7 + z^6 + 8*z^5 + z^4 + z^3 + z^2 + 7*zq^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^106323q^{2} + z^34940q^{3} + z^61064q^{4} + z^16824q^{5} + z^141263q^{6} + z^83787q^{7} + z^143207q^{8} + z^103307q^{9} + z^123147q^{10} + z^5872q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^155178)\omega^{1}&0\\0&u(z^5872)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(21387163858956230833 + O(11^20))*a^4 - (59404096813310740182 + O(11^20))*a^3 + (285434393449487570636 + O(11^20))*a^2 + (258382076893934702347 + O(11^20))*a + 325427090450479554154 + O(11^20)q^{3} + a^2 - 8192q^{4} + (149048911017397375406 + O(11^20))*a^4 - (212751438041340910991 + O(11^20))*a^3 + (169062434483015614325 + O(11^20))*a^2 - (93869867883927977009 + O(11^20))*a + 252179276043442089627 + O(11^20)q^{5} + (245758664138838808658 + O(11^20))*a^4 + (127053753102524643367 + O(11^20))*a^3 + (121601149095455710971 + O(11^20))*a^2 + (138425283053563602671 + O(11^20))*a + 212168488396872087078 + O(11^20)q^{6} + (216634659326732592466 + O(11^20))*a^4 + (17173076027588612481*11 + O(11^20))*a^3 - (79383888001421231121 + O(11^20))*a^2 - (168778358570009154338 + O(11^20))*a - 274123072264624336910 + O(11^20)q^{7} + a^3 - 16384*aq^{8} + -(292340748728231565075 + O(11^20))*a^4 - (311822893053470544917 + O(11^20))*a^3 - (157885957550417846730 + O(11^20))*a^2 + (228415951778337570761 + O(11^20))*a - 49953877595870578364 + O(11^20)q^{9} + -(19312600920288775336 + O(11^20))*a^4 - (131022188173597099255 + O(11^20))*a^3 + (273898172937446128592 + O(11^20))*a^2 - (58808135793201766318 + O(11^20))*a + 47512973853813980874 + O(11^20)q^{10} + (12633064790096194464*11 + O(11^20))*a^4 - (25533115118981745577*11 + O(11^20))*a^3 + (28779378768876882704*11 + O(11^20))*a^2 - (20699063029377833440*11 + O(11^20))*a - 24144466969759816095*11 + O(11^20)q^{11} + -(63849125614334263838 + O(11^20))*a^4 + (226173802006618836353 + O(11^20))*a^3 - (38968999222507754879 + O(11^20))*a^2 - (333800080229330717099 + O(11^20))*a - 1967432049382362515 + O(11^20)q^{12} + 4826809q^{13} + -(293667292172346286834 + O(11^20))*a^4 + (25674949162002874488 + O(11^20))*a^3 - (48914422817466958908 + O(11^20))*a^2 - (187333631007220538067 + O(11^20))*a + 164801161638778764219 + O(11^20)q^{14} + (180266126201550111797 + O(11^20))*a^4 - (14235042070596246561 + O(11^20))*a^3 - (273381884940763051701 + O(11^20))*a^2 + (175548833054252498936 + O(11^20))*a - 117119760516970272401 + O(11^20)q^{15} + a^4 - 24576*a^2 + 67108864q^{16} + -(263085504903913715282 + O(11^20))*a^4 + (86811378675964927121 + O(11^20))*a^3 + (59772151561100280682 + O(11^20))*a^2 - (278490724226963341212 + O(11^20))*a + 89678309788211630767 + O(11^20)q^{17} + (292960356135929954295 + O(11^20))*a^4 + (126788244543683647480 + O(11^20))*a^3 + (269630641206179273049 + O(11^20))*a^2 + (63933904165625109292 + O(11^20))*a - 40656570550301511864 + O(11^20)q^{18} + -(129342595797067967202 + O(11^20))*a^4 + (92833397036767088011 + O(11^20))*a^3 - (153708565622632542186 + O(11^20))*a^2 - (131759468696263651596 + O(11^20))*a + 208519509260609272666 + O(11^20)q^{19} + (155132964924173791298 + O(11^20))*a^4 + (164927169563368200405 + O(11^20))*a^3 - (207290028010266993472 + O(11^20))*a^2 + (88270279220978549808 + O(11^20))*a + 19657352748313031426 + O(11^20)q^{20} + -(95906054232732768975 + O(11^20))*a^4 + (66408531782613740267 + O(11^20))*a^3 + (143191402289721066832 + O(11^20))*a^2 + (141901363757969604243 + O(11^20))*a - 102563181908536027402 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -18405356653286492917 + O(11^20)x^{4} + -45435746779326533209 + O(11^20)x^{3} + 197200980531799123243 + O(11^20)x^{2} + -64509515765869694494 + O(11^20)x + -76188913029531034323 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9*z^9 + 7*z^8 + 10*z^7 + 10*z^6 + 6*z^5 + 6*z^4 + 2*z^3 + 4*z^2 + 6*zq^{2} + 9*z^9 + 2*z^8 + 7*z^7 + 6*z^6 + 5*z^5 + 4*z^4 + 5*z^3 + 4*z^2 + 5*zq^{3} + 9*z^9 + 7*z^8 + 3*z^7 + 5*z^6 + 3*z^5 + 3*z^4 + 7*z^3 + 3*z^2 + 3*z + 7q^{4} + 2*z^9 + 4*z^8 + 5*z^7 + 5*z^6 + 4*z^5 + 10*z^4 + 7*z^3 + 4*z^2 + 7*z + 10q^{5} + 10*z^9 + 3*z^8 + 4*z^7 + 6*z^6 + 4*z^5 + 7*z^4 + 9*z^3 + 6*z^2 + 5*z + 2q^{6} + 5*z^9 + 9*z^8 + 9*z^7 + 4*z^6 + 9*z^5 + 8*z^4 + 6*z^2 + 2*z + 10q^{7} + 7*z^9 + 7*z^7 + z^6 + 2*z^5 + 3*z^4 + 9*z^3 + 5*z + 10q^{8} + 4*z^9 + z^8 + z^7 + 3*z^6 + 2*z^4 + 5*z^3 + 4*z^2 + 5*z + 3q^{9} + 7*z^9 + z^8 + 5*z^7 + 10*z^6 + 8*z^5 + 7*z^4 + 9*z^3 + 10*z^2 + 9q^{10} + 9*z^9 + 5*z^8 + 5*z^7 + 5*z^6 + 10*z^5 + 6*z^4 + z^3 + 10*z^2 + 6*z + 4q^{12} + 9q^{13} + z^9 + 9*z^8 + 5*z^7 + 8*z^6 + 8*z^5 + 10*z^4 + 3*z^3 + 9*z^2 + 5*z + 5q^{14} + 10*z^9 + 2*z^8 + 2*z^6 + 5*z^5 + z^4 + z^3 + 6*z^2 + 7*z + 5q^{15} + 2*z^9 + z^8 + 10*z^7 + 9*z^6 + z^5 + 9*z^4 + 4*z^3 + 4*z^2 + 2*z + 7q^{16} + 3*z^9 + 9*z^8 + 7*z^7 + 4*z^6 + 5*z^5 + 6*z^4 + 3*zq^{17} + 2*z^9 + 8*z^6 + 9*z^4 + 3*z^3 + z^2 + 6*z + 2q^{18} + 2*z^9 + 3*z^8 + 6*z^6 + 2*z^5 + 9*z^4 + 4*z^3 + 8*z^2 + 10*z + 7q^{19} + 3*z^9 + 10*z^8 + 7*z^6 + 5*z^5 + 5*z^4 + 10*z^3 + 10*z^2 + 9*z + 7q^{20} + 6*z^9 + 3*z^7 + 2*z^6 + 5*z^5 + 7*z^4 + 8*z^3 + 6*z^2 + 5*zq^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^42203q^{2} + z^62240q^{3} + z^27504q^{4} + z^24014q^{5} + z^104443q^{6} + z^116407q^{7} + z^125827q^{8} + z^9027q^{9} + z^66217q^{10} + z^64592q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^96458)\omega^{1}&0\\0&u(z^64592)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(21387163858956230833 + O(11^20))*a^4 - (59404096813310740182 + O(11^20))*a^3 + (285434393449487570636 + O(11^20))*a^2 + (258382076893934702347 + O(11^20))*a + 325427090450479554154 + O(11^20)q^{3} + a^2 - 8192q^{4} + (149048911017397375406 + O(11^20))*a^4 - (212751438041340910991 + O(11^20))*a^3 + (169062434483015614325 + O(11^20))*a^2 - (93869867883927977009 + O(11^20))*a + 252179276043442089627 + O(11^20)q^{5} + (245758664138838808658 + O(11^20))*a^4 + (127053753102524643367 + O(11^20))*a^3 + (121601149095455710971 + O(11^20))*a^2 + (138425283053563602671 + O(11^20))*a + 212168488396872087078 + O(11^20)q^{6} + (216634659326732592466 + O(11^20))*a^4 + (17173076027588612481*11 + O(11^20))*a^3 - (79383888001421231121 + O(11^20))*a^2 - (168778358570009154338 + O(11^20))*a - 274123072264624336910 + O(11^20)q^{7} + a^3 - 16384*aq^{8} + -(292340748728231565075 + O(11^20))*a^4 - (311822893053470544917 + O(11^20))*a^3 - (157885957550417846730 + O(11^20))*a^2 + (228415951778337570761 + O(11^20))*a - 49953877595870578364 + O(11^20)q^{9} + -(19312600920288775336 + O(11^20))*a^4 - (131022188173597099255 + O(11^20))*a^3 + (273898172937446128592 + O(11^20))*a^2 - (58808135793201766318 + O(11^20))*a + 47512973853813980874 + O(11^20)q^{10} + (12633064790096194464*11 + O(11^20))*a^4 - (25533115118981745577*11 + O(11^20))*a^3 + (28779378768876882704*11 + O(11^20))*a^2 - (20699063029377833440*11 + O(11^20))*a - 24144466969759816095*11 + O(11^20)q^{11} + -(63849125614334263838 + O(11^20))*a^4 + (226173802006618836353 + O(11^20))*a^3 - (38968999222507754879 + O(11^20))*a^2 - (333800080229330717099 + O(11^20))*a - 1967432049382362515 + O(11^20)q^{12} + 4826809q^{13} + -(293667292172346286834 + O(11^20))*a^4 + (25674949162002874488 + O(11^20))*a^3 - (48914422817466958908 + O(11^20))*a^2 - (187333631007220538067 + O(11^20))*a + 164801161638778764219 + O(11^20)q^{14} + (180266126201550111797 + O(11^20))*a^4 - (14235042070596246561 + O(11^20))*a^3 - (273381884940763051701 + O(11^20))*a^2 + (175548833054252498936 + O(11^20))*a - 117119760516970272401 + O(11^20)q^{15} + a^4 - 24576*a^2 + 67108864q^{16} + -(263085504903913715282 + O(11^20))*a^4 + (86811378675964927121 + O(11^20))*a^3 + (59772151561100280682 + O(11^20))*a^2 - (278490724226963341212 + O(11^20))*a + 89678309788211630767 + O(11^20)q^{17} + (292960356135929954295 + O(11^20))*a^4 + (126788244543683647480 + O(11^20))*a^3 + (269630641206179273049 + O(11^20))*a^2 + (63933904165625109292 + O(11^20))*a - 40656570550301511864 + O(11^20)q^{18} + -(129342595797067967202 + O(11^20))*a^4 + (92833397036767088011 + O(11^20))*a^3 - (153708565622632542186 + O(11^20))*a^2 - (131759468696263651596 + O(11^20))*a + 208519509260609272666 + O(11^20)q^{19} + (155132964924173791298 + O(11^20))*a^4 + (164927169563368200405 + O(11^20))*a^3 - (207290028010266993472 + O(11^20))*a^2 + (88270279220978549808 + O(11^20))*a + 19657352748313031426 + O(11^20)q^{20} + -(95906054232732768975 + O(11^20))*a^4 + (66408531782613740267 + O(11^20))*a^3 + (143191402289721066832 + O(11^20))*a^2 + (141901363757969604243 + O(11^20))*a - 102563181908536027402 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -18405356653286492917 + O(11^20)x^{4} + -45435746779326533209 + O(11^20)x^{3} + 197200980531799123243 + O(11^20)x^{2} + -64509515765869694494 + O(11^20)x + -76188913029531034323 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10*z^9 + 10*z^8 + 9*z^7 + 5*z^6 + 6*z^5 + 3*z^4 + 6*z^3 + 9*z^2 + 2q^{2} + 10*z^9 + 4*z^8 + 5*z^7 + 6*z^6 + 2*z^5 + 5*z^4 + 4*z^3 + 10*z^2 + 4*z + 7q^{3} + 8*z^9 + 4*z^8 + 5*z^7 + 9*z^6 + 2*z^5 + 7*z^4 + z^3 + 9*z^2 + 9*z + 7q^{4} + 8*z^9 + z^8 + 7*z^7 + 7*z^5 + z^4 + 3*z^3 + 4*z^2 + 1q^{5} + 10*z^9 + 8*z^8 + 10*z^7 + 7*z^6 + 2*z^5 + z^4 + 6*z^2 + 6*z + 2q^{6} + 6*z^9 + 8*z^8 + 8*z^7 + 2*z^6 + 6*z^5 + 2*z^3 + 2*z^2 + 10*zq^{7} + 6*z^9 + 4*z^8 + 5*z^7 + 7*z^6 + 7*z^5 + 7*z^4 + 8*z^3 + 3*z^2 + 10*zq^{8} + z^8 + 8*z^7 + 8*z^6 + 9*z^5 + 6*z^3 + 8*z^2 + 10q^{9} + 4*z^9 + 8*z^8 + 4*z^7 + 2*z^6 + 6*z^5 + 4*z^4 + 10*z^3 + 5*z^2 + 5*z + 7q^{10} + z^9 + 2*z^8 + 8*z^7 + z^6 + 9*z^5 + z^3 + 6*z^2 + 10*z + 3q^{12} + 9q^{13} + 4*z^9 + 3*z^8 + 5*z^7 + 2*z^6 + 4*z^5 + 7*z^4 + 2*z^2 + 6*z + 9q^{14} + 7*z^9 + 4*z^8 + 9*z^7 + 10*z^6 + 9*z^4 + z^3 + z^2 + 2q^{15} + 3*z^9 + 3*z^8 + 5*z^7 + 6*z^6 + 7*z^5 + 9*z^4 + 10*z^3 + 4*z^2 + 5*z + 9q^{16} + 3*z^9 + 6*z^8 + 10*z^7 + 3*z^6 + 9*z^4 + 4*z^3 + 4*z^2 + 10*z + 3q^{17} + 4*z^9 + 5*z^8 + 8*z^6 + 8*z^5 + 6*z^3 + 9*z^2 + z + 1q^{18} + z^9 + z^8 + 8*z^7 + 7*z^6 + 3*z^5 + 8*z^4 + z^3 + 9*z^2 + 10*z + 9q^{19} + 5*z^9 + 2*z^8 + 10*z^7 + 6*z^6 + 5*z^5 + z^4 + 10*z^3 + 10*z^2 + 6*z + 5q^{20} + 6*z^9 + 2*z^8 + 9*z^7 + 10*z^6 + 7*z^5 + 3*z^4 + 10*z^3 + 4*z^2 + 6*z + 9q^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^126793q^{2} + z^61740q^{3} + z^49474q^{4} + z^74734q^{5} + z^27483q^{6} + z^7617q^{7} + z^144787q^{8} + z^97237q^{9} + z^40477q^{10} + z^132302q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^28748)\omega^{1}&0\\0&u(z^132302)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(21387163858956230833 + O(11^20))*a^4 - (59404096813310740182 + O(11^20))*a^3 + (285434393449487570636 + O(11^20))*a^2 + (258382076893934702347 + O(11^20))*a + 325427090450479554154 + O(11^20)q^{3} + a^2 - 8192q^{4} + (149048911017397375406 + O(11^20))*a^4 - (212751438041340910991 + O(11^20))*a^3 + (169062434483015614325 + O(11^20))*a^2 - (93869867883927977009 + O(11^20))*a + 252179276043442089627 + O(11^20)q^{5} + (245758664138838808658 + O(11^20))*a^4 + (127053753102524643367 + O(11^20))*a^3 + (121601149095455710971 + O(11^20))*a^2 + (138425283053563602671 + O(11^20))*a + 212168488396872087078 + O(11^20)q^{6} + (216634659326732592466 + O(11^20))*a^4 + (17173076027588612481*11 + O(11^20))*a^3 - (79383888001421231121 + O(11^20))*a^2 - (168778358570009154338 + O(11^20))*a - 274123072264624336910 + O(11^20)q^{7} + a^3 - 16384*aq^{8} + -(292340748728231565075 + O(11^20))*a^4 - (311822893053470544917 + O(11^20))*a^3 - (157885957550417846730 + O(11^20))*a^2 + (228415951778337570761 + O(11^20))*a - 49953877595870578364 + O(11^20)q^{9} + -(19312600920288775336 + O(11^20))*a^4 - (131022188173597099255 + O(11^20))*a^3 + (273898172937446128592 + O(11^20))*a^2 - (58808135793201766318 + O(11^20))*a + 47512973853813980874 + O(11^20)q^{10} + (12633064790096194464*11 + O(11^20))*a^4 - (25533115118981745577*11 + O(11^20))*a^3 + (28779378768876882704*11 + O(11^20))*a^2 - (20699063029377833440*11 + O(11^20))*a - 24144466969759816095*11 + O(11^20)q^{11} + -(63849125614334263838 + O(11^20))*a^4 + (226173802006618836353 + O(11^20))*a^3 - (38968999222507754879 + O(11^20))*a^2 - (333800080229330717099 + O(11^20))*a - 1967432049382362515 + O(11^20)q^{12} + 4826809q^{13} + -(293667292172346286834 + O(11^20))*a^4 + (25674949162002874488 + O(11^20))*a^3 - (48914422817466958908 + O(11^20))*a^2 - (187333631007220538067 + O(11^20))*a + 164801161638778764219 + O(11^20)q^{14} + (180266126201550111797 + O(11^20))*a^4 - (14235042070596246561 + O(11^20))*a^3 - (273381884940763051701 + O(11^20))*a^2 + (175548833054252498936 + O(11^20))*a - 117119760516970272401 + O(11^20)q^{15} + a^4 - 24576*a^2 + 67108864q^{16} + -(263085504903913715282 + O(11^20))*a^4 + (86811378675964927121 + O(11^20))*a^3 + (59772151561100280682 + O(11^20))*a^2 - (278490724226963341212 + O(11^20))*a + 89678309788211630767 + O(11^20)q^{17} + (292960356135929954295 + O(11^20))*a^4 + (126788244543683647480 + O(11^20))*a^3 + (269630641206179273049 + O(11^20))*a^2 + (63933904165625109292 + O(11^20))*a - 40656570550301511864 + O(11^20)q^{18} + -(129342595797067967202 + O(11^20))*a^4 + (92833397036767088011 + O(11^20))*a^3 - (153708565622632542186 + O(11^20))*a^2 - (131759468696263651596 + O(11^20))*a + 208519509260609272666 + O(11^20)q^{19} + (155132964924173791298 + O(11^20))*a^4 + (164927169563368200405 + O(11^20))*a^3 - (207290028010266993472 + O(11^20))*a^2 + (88270279220978549808 + O(11^20))*a + 19657352748313031426 + O(11^20)q^{20} + -(95906054232732768975 + O(11^20))*a^4 + (66408531782613740267 + O(11^20))*a^3 + (143191402289721066832 + O(11^20))*a^2 + (141901363757969604243 + O(11^20))*a - 102563181908536027402 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -18405356653286492917 + O(11^20)x^{4} + -45435746779326533209 + O(11^20)x^{3} + 197200980531799123243 + O(11^20)x^{2} + -64509515765869694494 + O(11^20)x + -76188913029531034323 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10*z^9 + 7*z^8 + 2*z^7 + 6*z^6 + 8*z^5 + 6*z^4 + 2*z^3 + 5*z^2 + 5*z + 1q^{2} + 10*z^9 + 4*z^8 + 8*z^7 + 6*z^6 + 7*z^5 + 7*z^4 + 3*z^3 + 2*z^2 + 2*z + 7q^{3} + 7*z^9 + 3*z^8 + 6*z^7 + 3*z^6 + 2*z^5 + z^4 + 6*z^3 + 10*z^2 + 7*z + 8q^{4} + 6*z^9 + 4*z^8 + 3*z^7 + 3*z^6 + 4*z^5 + 7*z^4 + 2*z^3 + 8*z^2 + 3*z + 8q^{5} + 7*z^9 + 10*z^8 + 6*z^7 + 10*z^6 + 6*z^5 + 2*z^4 + 7*z^3 + 5*z^2 + 9q^{6} + 5*z^9 + 8*z^8 + 5*z^7 + 2*z^6 + 8*z^5 + 5*z^4 + 10*z^3 + 8*z^2 + 10*z + 7q^{7} + 10*z^9 + z^8 + 4*z^7 + 6*z^6 + 4*z^5 + 5*z^4 + 2*z + 8q^{8} + 10*z^9 + 8*z^8 + 10*z^7 + 10*z^6 + 10*z^5 + 5*z^3 + 9*z^2 + 4*z + 7q^{9} + 7*z^9 + 2*z^8 + 9*z^7 + 3*z^6 + 7*z^5 + 2*z^4 + 6*z^3 + 10*z^2 + 9*z + 8q^{10} + 2*z^9 + 5*z^8 + 8*z^5 + 2*z^4 + 3*z^3 + 8*z^2 + 6*z + 8q^{12} + 9q^{13} + 3*z^9 + 3*z^8 + 2*z^7 + z^6 + 7*z^5 + 5*z^4 + 10*z^3 + 7*z^2 + 3*z + 7q^{14} + 8*z^9 + 9*z^8 + 10*z^7 + 5*z^6 + 7*z^5 + 2*z^4 + 6*z^3 + 3*z^2 + 3*z + 7q^{15} + 4*z^9 + 3*z^8 + z^7 + 2*z^6 + z^5 + 8*z^4 + 5*z^3 + 9*z^2 + 5*z + 10q^{16} + z^9 + 4*z^8 + 10*z^7 + 8*z^6 + 9*z^5 + z^4 + 10*z^3 + 10*z^2 + 7*z + 4q^{17} + 6*z^8 + 4*z^7 + z^6 + 2*z^5 + z^4 + 9*z^3 + z^2 + 2*z + 6q^{18} + z^9 + 2*z^8 + z^6 + 3*z^5 + 2*z^4 + 9*z^2 + 8*z + 6q^{19} + 4*z^9 + 7*z^8 + z^7 + 8*z^6 + 6*z^5 + 10*z^4 + 5*z^2 + 7*z + 10q^{20} + 2*z^9 + 2*z^8 + 9*z^7 + 3*z^6 + 9*z^5 + 4*z^4 + 8*z^3 + 10*z + 7q^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^114013q^{2} + z^122740q^{3} + z^106984q^{4} + z^6794q^{5} + z^75703q^{6} + z^73897q^{7} + z^86367q^{8} + z^125967q^{9} + z^120807q^{10} + z^85232q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^75818)\omega^{1}&0\\0&u(z^85232)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(21387163858956230833 + O(11^20))*a^4 - (59404096813310740182 + O(11^20))*a^3 + (285434393449487570636 + O(11^20))*a^2 + (258382076893934702347 + O(11^20))*a + 325427090450479554154 + O(11^20)q^{3} + a^2 - 8192q^{4} + (149048911017397375406 + O(11^20))*a^4 - (212751438041340910991 + O(11^20))*a^3 + (169062434483015614325 + O(11^20))*a^2 - (93869867883927977009 + O(11^20))*a + 252179276043442089627 + O(11^20)q^{5} + (245758664138838808658 + O(11^20))*a^4 + (127053753102524643367 + O(11^20))*a^3 + (121601149095455710971 + O(11^20))*a^2 + (138425283053563602671 + O(11^20))*a + 212168488396872087078 + O(11^20)q^{6} + (216634659326732592466 + O(11^20))*a^4 + (17173076027588612481*11 + O(11^20))*a^3 - (79383888001421231121 + O(11^20))*a^2 - (168778358570009154338 + O(11^20))*a - 274123072264624336910 + O(11^20)q^{7} + a^3 - 16384*aq^{8} + -(292340748728231565075 + O(11^20))*a^4 - (311822893053470544917 + O(11^20))*a^3 - (157885957550417846730 + O(11^20))*a^2 + (228415951778337570761 + O(11^20))*a - 49953877595870578364 + O(11^20)q^{9} + -(19312600920288775336 + O(11^20))*a^4 - (131022188173597099255 + O(11^20))*a^3 + (273898172937446128592 + O(11^20))*a^2 - (58808135793201766318 + O(11^20))*a + 47512973853813980874 + O(11^20)q^{10} + (12633064790096194464*11 + O(11^20))*a^4 - (25533115118981745577*11 + O(11^20))*a^3 + (28779378768876882704*11 + O(11^20))*a^2 - (20699063029377833440*11 + O(11^20))*a - 24144466969759816095*11 + O(11^20)q^{11} + -(63849125614334263838 + O(11^20))*a^4 + (226173802006618836353 + O(11^20))*a^3 - (38968999222507754879 + O(11^20))*a^2 - (333800080229330717099 + O(11^20))*a - 1967432049382362515 + O(11^20)q^{12} + 4826809q^{13} + -(293667292172346286834 + O(11^20))*a^4 + (25674949162002874488 + O(11^20))*a^3 - (48914422817466958908 + O(11^20))*a^2 - (187333631007220538067 + O(11^20))*a + 164801161638778764219 + O(11^20)q^{14} + (180266126201550111797 + O(11^20))*a^4 - (14235042070596246561 + O(11^20))*a^3 - (273381884940763051701 + O(11^20))*a^2 + (175548833054252498936 + O(11^20))*a - 117119760516970272401 + O(11^20)q^{15} + a^4 - 24576*a^2 + 67108864q^{16} + -(263085504903913715282 + O(11^20))*a^4 + (86811378675964927121 + O(11^20))*a^3 + (59772151561100280682 + O(11^20))*a^2 - (278490724226963341212 + O(11^20))*a + 89678309788211630767 + O(11^20)q^{17} + (292960356135929954295 + O(11^20))*a^4 + (126788244543683647480 + O(11^20))*a^3 + (269630641206179273049 + O(11^20))*a^2 + (63933904165625109292 + O(11^20))*a - 40656570550301511864 + O(11^20)q^{18} + -(129342595797067967202 + O(11^20))*a^4 + (92833397036767088011 + O(11^20))*a^3 - (153708565622632542186 + O(11^20))*a^2 - (131759468696263651596 + O(11^20))*a + 208519509260609272666 + O(11^20)q^{19} + (155132964924173791298 + O(11^20))*a^4 + (164927169563368200405 + O(11^20))*a^3 - (207290028010266993472 + O(11^20))*a^2 + (88270279220978549808 + O(11^20))*a + 19657352748313031426 + O(11^20)q^{20} + -(95906054232732768975 + O(11^20))*a^4 + (66408531782613740267 + O(11^20))*a^3 + (143191402289721066832 + O(11^20))*a^2 + (141901363757969604243 + O(11^20))*a - 102563181908536027402 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + -18405356653286492917 + O(11^20)x^{4} + -45435746779326533209 + O(11^20)x^{3} + 197200980531799123243 + O(11^20)x^{2} + -64509515765869694494 + O(11^20)x + -76188913029531034323 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^67q^{2} + z^4q^{3} + z^63q^{4} + q^{5} + z^71q^{6} + z^69q^{7} + 5q^{8} + z^26q^{9} + z^67q^{10} + z^67q^{12} + 7q^{13} + z^16q^{14} + z^4q^{15} + z^81q^{16} + z^107q^{17} + z^93q^{18} + z^101q^{19} + z^63q^{20} + z^73q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^19q^{2} + z^100q^{3} + z^87q^{4} + 5q^{5} + z^119q^{6} + z^93q^{7} + 4q^{8} + z^98q^{9} + z^67q^{10} + z^50q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^70)\omega^{1}&0\\0&u(z^50)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (25415011363644582209 + O(11^20))*a - 184760342222364118234 + O(11^20)q^{3} + -(335474258096283468675 + O(11^20))*a - 177645760487242438617 + O(11^20)q^{4} + -(2538309410536586017*11 + O(11^20))*a - 81830498585879039578 + O(11^20)q^{5} + (331673390413669917036 + O(11^20))*a - 169715914428692509177 + O(11^20)q^{6} + (50648189589551431646 + O(11^20))*a + 186827113208490646372 + O(11^20)q^{7} + -(3428561964719376846*11 + O(11^20))*a - 126226405244072606787 + O(11^20)q^{8} + (219078988456052067868 + O(11^20))*a - 243181394710002307672 + O(11^20)q^{9} + -(272198639486647973593 + O(11^20))*a - 19850484618106774163*11 + O(11^20)q^{10} + (14781856637584693792*11 + O(11^20))*a + 12653974711060250344*11 + O(11^20)q^{11} + (323315772427642468662 + O(11^20))*a - 9945905551271317409*11 + O(11^20)q^{12} + 62748517q^{13} + (210111756423323978051 + O(11^20))*a + 179351936869711050034 + O(11^20)q^{14} + (321803059274870130355 + O(11^20))*a - 86336895966490210604 + O(11^20)q^{15} + -(300751578085830773805 + O(11^20))*a + 286334523221002952547 + O(11^20)q^{16} + (11607967999906930832 + O(11^20))*a + 215584749729566945000 + O(11^20)q^{17} + (226592800325333284729 + O(11^20))*a + 336370467436025007270 + O(11^20)q^{18} + (251460991733816605514 + O(11^20))*a + 296513207093114043005 + O(11^20)q^{19} + (233611048864643133413 + O(11^20))*a + 178143539783441842043 + O(11^20)q^{20} + (260457718896221044318 + O(11^20))*a + 88069538065003986005 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 335474258096283468675 + O(11^20)x + 177645760487242405849 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 5q^{3} + 4q^{4} + 7q^{5} + 8q^{6} + 5q^{7} + 8q^{8} + 2q^{9} + 9q^{10} + 9q^{12} + 7q^{13} + 8q^{14} + 2q^{15} + 8q^{16} + q^{18} + 5q^{19} + 6q^{20} + 3q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 10q^{2} + 4q^{3} + 5q^{4} + 2q^{5} + 7q^{6} + 9q^{7} + 2q^{8} + 7q^{9} + 9q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(10)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 173844857799179963336 + O(11^20)q^{2} + 240073973805460162615 + O(11^20)q^{3} + -335724342202001875437 + O(11^20)q^{4} + 80898634712599570588 + O(11^20)q^{5} + 287337641378173936041 + O(11^20)q^{6} + 237598086791698877491 + O(11^20)q^{7} + 96450322999256137997 + O(11^20)q^{8} + 106100770004383399401 + O(11^20)q^{9} + -100043751548761707491 + O(11^20)q^{10} + 2997822834763267387*11 + O(11^20)q^{11} + 15015683595807241136 + O(11^20)q^{12} + 62748517q^{13} + -293988603175990225715 + O(11^20)q^{14} + 203320461554941773352 + O(11^20)q^{15} + 276666081118969051221 + O(11^20)q^{16} + -9683728259889591612*11 + O(11^20)q^{17} + 168731992359328732621 + O(11^20)q^{18} + -50727812382215600154 + O(11^20)q^{19} + -252020560701904513469 + O(11^20)q^{20} + 198854047385826329786 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^109q^{2} + z^46q^{3} + z^117q^{4} + z^104q^{5} + z^35q^{6} + 7q^{7} + 10q^{8} + z^77q^{9} + z^93q^{10} + z^43q^{12} + 7q^{13} + z^73q^{14} + z^30q^{15} + z^15q^{16} + z^67q^{17} + z^66q^{18} + z^9q^{19} + z^101q^{20} + z^10q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^97q^{2} + z^70q^{3} + z^93q^{4} + z^56q^{5} + z^47q^{6} + q^{7} + 4q^{8} + z^5q^{9} + z^33q^{10} + z^76q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^44)\omega^{4}&0\\0&u(z^76)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (24237401115059786149 + O(11^20))*a + 292128984187267580101 + O(11^20)q^{3} + -(169869830449523554579 + O(11^20))*a + 264330951694383892236 + O(11^20)q^{4} + -(186494590566620281773 + O(11^20))*a - 113805390928469554456 + O(11^20)q^{5} + (145288154855509583030 + O(11^20))*a + 69621407390455736942 + O(11^20)q^{6} + (9460327178388909489*11 + O(11^20))*a - 32314192975689651338 + O(11^20)q^{7} + (29867268918349018364*11 + O(11^20))*a - 232169834773474077547 + O(11^20)q^{8} + (315777730859310409232 + O(11^20))*a - 72808936885031049852 + O(11^20)q^{9} + (118308611458871685165 + O(11^20))*a + 139262634905023607831 + O(11^20)q^{10} + (5435952445308691654*11 + O(11^20))*a + 7647174326962571591*11 + O(11^20)q^{11} + -(204025216163668439451 + O(11^20))*a + 64457689156898983676 + O(11^20)q^{12} + 62748517q^{13} + (289802594587122485678 + O(11^20))*a + 17043740203554914313*11 + O(11^20)q^{14} + -(52194490642369756590 + O(11^20))*a + 295389411729825687425 + O(11^20)q^{15} + -(237545632949919238867 + O(11^20))*a - 148283632252491498834 + O(11^20)q^{16} + -(191713586535369158025 + O(11^20))*a - 204377688102865711326 + O(11^20)q^{17} + -(184928893141485477532 + O(11^20))*a - 149760491053600479538 + O(11^20)q^{18} + -(200451390481843420982 + O(11^20))*a + 131622306480198700706 + O(11^20)q^{19} + -(38293811984011086344 + O(11^20))*a + 109604457245176214781 + O(11^20)q^{20} + -(196626535484987889527 + O(11^20))*a - 214888379299636815975 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 169869830449523554579 + O(11^20)x + -264330951694383925004 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^119q^{2} + z^26q^{3} + z^87q^{4} + z^64q^{5} + z^25q^{6} + 7q^{7} + 10q^{8} + z^7q^{9} + z^63q^{10} + z^113q^{12} + 7q^{13} + z^83q^{14} + z^90q^{15} + z^45q^{16} + z^17q^{17} + z^6q^{18} + z^99q^{19} + z^31q^{20} + z^110q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^107q^{2} + z^50q^{3} + z^63q^{4} + z^16q^{5} + z^37q^{6} + q^{7} + 4q^{8} + z^55q^{9} + z^3q^{10} + z^116q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^4)\omega^{4}&0\\0&u(z^116)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (24237401115059786149 + O(11^20))*a + 292128984187267580101 + O(11^20)q^{3} + -(169869830449523554579 + O(11^20))*a + 264330951694383892236 + O(11^20)q^{4} + -(186494590566620281773 + O(11^20))*a - 113805390928469554456 + O(11^20)q^{5} + (145288154855509583030 + O(11^20))*a + 69621407390455736942 + O(11^20)q^{6} + (9460327178388909489*11 + O(11^20))*a - 32314192975689651338 + O(11^20)q^{7} + (29867268918349018364*11 + O(11^20))*a - 232169834773474077547 + O(11^20)q^{8} + (315777730859310409232 + O(11^20))*a - 72808936885031049852 + O(11^20)q^{9} + (118308611458871685165 + O(11^20))*a + 139262634905023607831 + O(11^20)q^{10} + (5435952445308691654*11 + O(11^20))*a + 7647174326962571591*11 + O(11^20)q^{11} + -(204025216163668439451 + O(11^20))*a + 64457689156898983676 + O(11^20)q^{12} + 62748517q^{13} + (289802594587122485678 + O(11^20))*a + 17043740203554914313*11 + O(11^20)q^{14} + -(52194490642369756590 + O(11^20))*a + 295389411729825687425 + O(11^20)q^{15} + -(237545632949919238867 + O(11^20))*a - 148283632252491498834 + O(11^20)q^{16} + -(191713586535369158025 + O(11^20))*a - 204377688102865711326 + O(11^20)q^{17} + -(184928893141485477532 + O(11^20))*a - 149760491053600479538 + O(11^20)q^{18} + -(200451390481843420982 + O(11^20))*a + 131622306480198700706 + O(11^20)q^{19} + -(38293811984011086344 + O(11^20))*a + 109604457245176214781 + O(11^20)q^{20} + -(196626535484987889527 + O(11^20))*a - 214888379299636815975 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 169869830449523554579 + O(11^20)x + -264330951694383925004 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^17q^{2} + z^44q^{3} + z^93q^{4} + q^{5} + z^61q^{6} + z^39q^{7} + 5q^{8} + z^46q^{9} + z^17q^{10} + z^17q^{12} + 7q^{13} + z^56q^{14} + z^44q^{15} + z^51q^{16} + z^97q^{17} + z^63q^{18} + z^31q^{19} + z^93q^{20} + z^83q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^89q^{2} + z^20q^{3} + z^117q^{4} + 5q^{5} + z^109q^{6} + z^63q^{7} + 4q^{8} + z^118q^{9} + z^17q^{10} + z^70q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^50)\omega^{1}&0\\0&u(z^70)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (25415011363644582209 + O(11^20))*a - 184760342222364118234 + O(11^20)q^{3} + -(335474258096283468675 + O(11^20))*a - 177645760487242438617 + O(11^20)q^{4} + -(2538309410536586017*11 + O(11^20))*a - 81830498585879039578 + O(11^20)q^{5} + (331673390413669917036 + O(11^20))*a - 169715914428692509177 + O(11^20)q^{6} + (50648189589551431646 + O(11^20))*a + 186827113208490646372 + O(11^20)q^{7} + -(3428561964719376846*11 + O(11^20))*a - 126226405244072606787 + O(11^20)q^{8} + (219078988456052067868 + O(11^20))*a - 243181394710002307672 + O(11^20)q^{9} + -(272198639486647973593 + O(11^20))*a - 19850484618106774163*11 + O(11^20)q^{10} + (14781856637584693792*11 + O(11^20))*a + 12653974711060250344*11 + O(11^20)q^{11} + (323315772427642468662 + O(11^20))*a - 9945905551271317409*11 + O(11^20)q^{12} + 62748517q^{13} + (210111756423323978051 + O(11^20))*a + 179351936869711050034 + O(11^20)q^{14} + (321803059274870130355 + O(11^20))*a - 86336895966490210604 + O(11^20)q^{15} + -(300751578085830773805 + O(11^20))*a + 286334523221002952547 + O(11^20)q^{16} + (11607967999906930832 + O(11^20))*a + 215584749729566945000 + O(11^20)q^{17} + (226592800325333284729 + O(11^20))*a + 336370467436025007270 + O(11^20)q^{18} + (251460991733816605514 + O(11^20))*a + 296513207093114043005 + O(11^20)q^{19} + (233611048864643133413 + O(11^20))*a + 178143539783441842043 + O(11^20)q^{20} + (260457718896221044318 + O(11^20))*a + 88069538065003986005 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 335474258096283468675 + O(11^20)x + 177645760487242405849 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^774q^{2} + z^1104q^{3} + z^1256q^{4} + z^803q^{5} + z^548q^{6} + z^630q^{7} + z^282q^{8} + z^571q^{9} + z^247q^{10} + z^1030q^{12} + 4q^{13} + z^74q^{14} + z^577q^{15} + z^2q^{16} + z^1190q^{17} + z^15q^{18} + z^681q^{19} + z^729q^{20} + z^404q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^242q^{2} + z^838q^{3} + z^192q^{4} + z^5q^{5} + z^1080q^{6} + z^896q^{7} + z^16q^{8} + z^39q^{9} + z^247q^{10} + z^6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1324)\omega^{1}&0\\0&u(z^6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (107392270277215435395 + O(11^20))*a^2 - (22348708376847180899*11 + O(11^20))*a + 176311994708273597588 + O(11^20)q^{3} + a^2 - 32768q^{4} + (323453728731794453944 + O(11^20))*a^2 - (268631194550805867401 + O(11^20))*a - 122406606024983023102 + O(11^20)q^{5} + -(309116401493407520609 + O(11^20))*a^2 - (168628574716055392440 + O(11^20))*a - 73220034978376798151 + O(11^20)q^{6} + -(70232318676624297606 + O(11^20))*a^2 - (17773547368689671701 + O(11^20))*a + 243932781102322494862 + O(11^20)q^{7} + -(171610909919118756777 + O(11^20))*a^2 + (263757824938559686483 + O(11^20))*a - 218293975610444480577 + O(11^20)q^{8} + (248668873872975819408 + O(11^20))*a^2 - (104536185811181809923 + O(11^20))*a + 43493232672519273262 + O(11^20)q^{9} + (24940688280074130633*11 + O(11^20))*a^2 - (21592929638510053036 + O(11^20))*a + 295326816074655749094 + O(11^20)q^{10} + -(26809171886242918035*11 + O(11^20))*a^2 + (1404966355562807828*11 + O(11^20))*a - 25676307324502447280*11 + O(11^20)q^{11} + (260101884241989103767 + O(11^20))*a^2 + (302384616337052801014 + O(11^20))*a - 185204009850196427108 + O(11^20)q^{12} + -62748517q^{13} + (9826468108560345145*11 + O(11^20))*a^2 + (176926083828782911517 + O(11^20))*a - 318838286544913824400 + O(11^20)q^{14} + (175404692934669002270 + O(11^20))*a^2 + (156991144029098306333 + O(11^20))*a + 44264371734777735144 + O(11^20)q^{15} + -(278923165780171530073 + O(11^20))*a^2 + (144993764922377329060 + O(11^20))*a + 15382257327926584732*11 + O(11^20)q^{16} + -(96703876150915456301 + O(11^20))*a^2 + (13816540544235036757*11 + O(11^20))*a - 312282454989481189918 + O(11^20)q^{17} + -(78636032597657674494 + O(11^20))*a^2 + (291822973381490966306 + O(11^20))*a - 200834409958368412370 + O(11^20)q^{18} + (331337894734084397923 + O(11^20))*a^2 - (163878108839801259555 + O(11^20))*a - 90398420624017575018 + O(11^20)q^{19} + -(112149562426969479738 + O(11^20))*a^2 + (197186223990813612909 + O(11^20))*a - 208423537557244500872 + O(11^20)q^{20} + (150697258076683180512 + O(11^20))*a^2 - (85957143302074125930 + O(11^20))*a - 128066842017984476194 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 171610909919118756777 + O(11^20)x^{2} + -263757824938559752019 + O(11^20)x + 218293975610444480577 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 2q^{3} + 6q^{4} + 5q^{5} + 3q^{6} + 10q^{7} + 5q^{8} + 3q^{9} + 2q^{10} + q^{12} + 4q^{13} + 4q^{14} + 10q^{15} + 8q^{16} + q^{17} + 10q^{18} + 10q^{19} + 8q^{20} + 9q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 6q^{3} + 2q^{4} + 3q^{5} + 4q^{6} + 7q^{7} + 4q^{8} + 5q^{9} + 2q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(3)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 120017050706114314430 + O(11^20)q^{2} + 222386774395346154420 + O(11^20)q^{3} + -186179288618642499713 + O(11^20)q^{4} + -72488432383327953262 + O(11^20)q^{5} + -274853195572741581947 + O(11^20)q^{6} + -225508598224753970666 + O(11^20)q^{7} + 218715252251246777284 + O(11^20)q^{8} + -252571510513344599651 + O(11^20)q^{9} + 129916529166247941262 + O(11^20)q^{10} + 3584757830220004450*11 + O(11^20)q^{11} + 23944954636428774840 + O(11^20)q^{12} + -62748517q^{13} + 62972509679491795651 + O(11^20)q^{14} + 284222649833340466533 + O(11^20)q^{15} + -108955160217259638769 + O(11^20)q^{16} + 45307078439763058645 + O(11^20)q^{17} + 140119812713102471919 + O(11^20)q^{18} + 327622480922146838779 + O(11^20)q^{19} + 110236469566436168755 + O(11^20)q^{20} + 214150103376108056100 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{3} + 2q^{4} + 9q^{5} + q^{6} + 8q^{7} + 3q^{8} + 9q^{10} + 2q^{12} + 4q^{13} + 8q^{14} + 9q^{15} + 5q^{16} + 4q^{19} + 7q^{20} + 8q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 6q^{2} + 4q^{3} + 6q^{4} + 4q^{5} + 2q^{6} + 9q^{7} + 10q^{8} + 2q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{4}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -107754626022217073184 + O(11^20)q^{2} + -253497724721559854348 + O(11^20)q^{3} + -43024237534570777153 + O(11^20)q^{4} + -295364415211796863143 + O(11^20)q^{5} + 42312148037287521097 + O(11^20)q^{6} + -132578043065231485252 + O(11^20)q^{7} + 108285930612604850768 + O(11^20)q^{8} + -11025653945703364270*11 + O(11^20)q^{9} + 38254679475606266352 + O(11^20)q^{10} + -24010705195019425719*11 + O(11^20)q^{11} + 132950460521916639593 + O(11^20)q^{12} + -62748517q^{13} + 105573664999695019828 + O(11^20)q^{14} + 252835307699693179459 + O(11^20)q^{15} + -247014496824987730471 + O(11^20)q^{16} + -10677820931093042125*11 + O(11^20)q^{17} + -20833574399729661400*11 + O(11^20)q^{18} + -278889103086255131840 + O(11^20)q^{19} + -239110319563539755874 + O(11^20)q^{20} + -183357935691745192139 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^534q^{2} + z^174q^{3} + z^516q^{4} + z^853q^{5} + z^708q^{6} + z^280q^{7} + z^442q^{8} + z^961q^{9} + z^57q^{10} + z^690q^{12} + 4q^{13} + z^814q^{14} + z^1027q^{15} + z^22q^{16} + z^1120q^{17} + z^165q^{18} + z^841q^{19} + z^39q^{20} + z^454q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^2q^{2} + z^1238q^{3} + z^782q^{4} + z^55q^{5} + z^1240q^{6} + z^546q^{7} + z^176q^{8} + z^429q^{9} + z^57q^{10} + z^66q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1264)\omega^{1}&0\\0&u(z^66)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (107392270277215435395 + O(11^20))*a^2 - (22348708376847180899*11 + O(11^20))*a + 176311994708273597588 + O(11^20)q^{3} + a^2 - 32768q^{4} + (323453728731794453944 + O(11^20))*a^2 - (268631194550805867401 + O(11^20))*a - 122406606024983023102 + O(11^20)q^{5} + -(309116401493407520609 + O(11^20))*a^2 - (168628574716055392440 + O(11^20))*a - 73220034978376798151 + O(11^20)q^{6} + -(70232318676624297606 + O(11^20))*a^2 - (17773547368689671701 + O(11^20))*a + 243932781102322494862 + O(11^20)q^{7} + -(171610909919118756777 + O(11^20))*a^2 + (263757824938559686483 + O(11^20))*a - 218293975610444480577 + O(11^20)q^{8} + (248668873872975819408 + O(11^20))*a^2 - (104536185811181809923 + O(11^20))*a + 43493232672519273262 + O(11^20)q^{9} + (24940688280074130633*11 + O(11^20))*a^2 - (21592929638510053036 + O(11^20))*a + 295326816074655749094 + O(11^20)q^{10} + -(26809171886242918035*11 + O(11^20))*a^2 + (1404966355562807828*11 + O(11^20))*a - 25676307324502447280*11 + O(11^20)q^{11} + (260101884241989103767 + O(11^20))*a^2 + (302384616337052801014 + O(11^20))*a - 185204009850196427108 + O(11^20)q^{12} + -62748517q^{13} + (9826468108560345145*11 + O(11^20))*a^2 + (176926083828782911517 + O(11^20))*a - 318838286544913824400 + O(11^20)q^{14} + (175404692934669002270 + O(11^20))*a^2 + (156991144029098306333 + O(11^20))*a + 44264371734777735144 + O(11^20)q^{15} + -(278923165780171530073 + O(11^20))*a^2 + (144993764922377329060 + O(11^20))*a + 15382257327926584732*11 + O(11^20)q^{16} + -(96703876150915456301 + O(11^20))*a^2 + (13816540544235036757*11 + O(11^20))*a - 312282454989481189918 + O(11^20)q^{17} + -(78636032597657674494 + O(11^20))*a^2 + (291822973381490966306 + O(11^20))*a - 200834409958368412370 + O(11^20)q^{18} + (331337894734084397923 + O(11^20))*a^2 - (163878108839801259555 + O(11^20))*a - 90398420624017575018 + O(11^20)q^{19} + -(112149562426969479738 + O(11^20))*a^2 + (197186223990813612909 + O(11^20))*a - 208423537557244500872 + O(11^20)q^{20} + (150697258076683180512 + O(11^20))*a^2 - (85957143302074125930 + O(11^20))*a - 128066842017984476194 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 171610909919118756777 + O(11^20)x^{2} + -263757824938559752019 + O(11^20)x + 218293975610444480577 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^554q^{2} + z^584q^{3} + z^356q^{4} + z^73q^{5} + z^1138q^{6} + z^420q^{7} + z^872q^{8} + z^1261q^{9} + z^627q^{10} + z^940q^{12} + 4q^{13} + z^974q^{14} + z^657q^{15} + z^242q^{16} + z^350q^{17} + z^485q^{18} + z^1271q^{19} + z^429q^{20} + z^1004q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^22q^{2} + z^318q^{3} + z^622q^{4} + z^605q^{5} + z^340q^{6} + z^686q^{7} + z^606q^{8} + z^729q^{9} + z^627q^{10} + z^726q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^604)\omega^{1}&0\\0&u(z^726)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (107392270277215435395 + O(11^20))*a^2 - (22348708376847180899*11 + O(11^20))*a + 176311994708273597588 + O(11^20)q^{3} + a^2 - 32768q^{4} + (323453728731794453944 + O(11^20))*a^2 - (268631194550805867401 + O(11^20))*a - 122406606024983023102 + O(11^20)q^{5} + -(309116401493407520609 + O(11^20))*a^2 - (168628574716055392440 + O(11^20))*a - 73220034978376798151 + O(11^20)q^{6} + -(70232318676624297606 + O(11^20))*a^2 - (17773547368689671701 + O(11^20))*a + 243932781102322494862 + O(11^20)q^{7} + -(171610909919118756777 + O(11^20))*a^2 + (263757824938559686483 + O(11^20))*a - 218293975610444480577 + O(11^20)q^{8} + (248668873872975819408 + O(11^20))*a^2 - (104536185811181809923 + O(11^20))*a + 43493232672519273262 + O(11^20)q^{9} + (24940688280074130633*11 + O(11^20))*a^2 - (21592929638510053036 + O(11^20))*a + 295326816074655749094 + O(11^20)q^{10} + -(26809171886242918035*11 + O(11^20))*a^2 + (1404966355562807828*11 + O(11^20))*a - 25676307324502447280*11 + O(11^20)q^{11} + (260101884241989103767 + O(11^20))*a^2 + (302384616337052801014 + O(11^20))*a - 185204009850196427108 + O(11^20)q^{12} + -62748517q^{13} + (9826468108560345145*11 + O(11^20))*a^2 + (176926083828782911517 + O(11^20))*a - 318838286544913824400 + O(11^20)q^{14} + (175404692934669002270 + O(11^20))*a^2 + (156991144029098306333 + O(11^20))*a + 44264371734777735144 + O(11^20)q^{15} + -(278923165780171530073 + O(11^20))*a^2 + (144993764922377329060 + O(11^20))*a + 15382257327926584732*11 + O(11^20)q^{16} + -(96703876150915456301 + O(11^20))*a^2 + (13816540544235036757*11 + O(11^20))*a - 312282454989481189918 + O(11^20)q^{17} + -(78636032597657674494 + O(11^20))*a^2 + (291822973381490966306 + O(11^20))*a - 200834409958368412370 + O(11^20)q^{18} + (331337894734084397923 + O(11^20))*a^2 - (163878108839801259555 + O(11^20))*a - 90398420624017575018 + O(11^20)q^{19} + -(112149562426969479738 + O(11^20))*a^2 + (197186223990813612909 + O(11^20))*a - 208423537557244500872 + O(11^20)q^{20} + (150697258076683180512 + O(11^20))*a^2 - (85957143302074125930 + O(11^20))*a - 128066842017984476194 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 171610909919118756777 + O(11^20)x^{2} + -263757824938559752019 + O(11^20)x + 218293975610444480577 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10*z^5 + 8*z^4 + 8*z^3 + 5*z^2 + 3q^{2} + 4*z^5 + 7*z^4 + 7*z^3 + 9*z^2 + 5*z + 3q^{3} + 3*z^5 + 10*z^4 + 10*z^3 + 10*z^2 + 10*zq^{4} + 7*z^5 + 9*z^4 + 9*z^3 + 6*z^2 + zq^{5} + 9*z^5 + 4*z^4 + 4*z^3 + 7*z^2 + z + 7q^{6} + 6*z^5 + 5*z^4 + 5*z^3 + 8*z^2 + 2*z + 1q^{7} + 6*z^5 + 10*z^4 + 10*z^3 + z^2 + 8*z + 9q^{8} + 5*z^5 + 8*z^4 + 8*z^3 + 9*z^2 + 7*z + 2q^{9} + 7*z^5 + 8*z^4 + 8*z^3 + 3*z^2 + 2*z + 1q^{10} + 4*z^5 + 10*z^4 + 10*z^3 + 7*z^2 + 2*z + 5q^{12} + 8q^{13} + 6*z^5 + 4*z^2 + 7*zq^{14} + 4*z^5 + 6*z^4 + 6*z^3 + 6*z^2 + 6*z + 4q^{15} + 7*z^5 + 4*z^4 + 4*z^3 + 2*z^2 + 6*z + 1q^{16} + 3*z^5 + 3*z^4 + 3*z^3 + 6*z + 3q^{17} + 3*z^5 + 6*z^4 + 6*z^3 + 9*z^2 + 3*z + 2q^{18} + 9*z^5 + 4*z^4 + 4*z^3 + 7*z^2 + z + 4q^{19} + 6*z^5 + 6*z^4 + 6*z^3 + z + 2q^{20} + 10*z^5 + 4*z^4 + 4*z^3 + 4*z^2 + 4*z + 6q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^1097q^{2} + z^292q^{3} + z^1006q^{4} + z^1249q^{5} + z^59q^{6} + z^1311q^{7} + z^838q^{8} + z^616q^{9} + z^1016q^{10} + z^367q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^963)\omega^{6}&0\\0&u(z^367)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(131446899915469254352 + O(11^20))*a^2 - (16668102967084971608 + O(11^20))*a - 143028618626631968157 + O(11^20)q^{3} + a^2 - 131072q^{4} + (248431489477194627658 + O(11^20))*a^2 - (62985048310559850397 + O(11^20))*a - 107124721210528445277 + O(11^20)q^{5} + -(112777766827640977646 + O(11^20))*a^2 + (115594248386970322716 + O(11^20))*a - 65634966030451128723 + O(11^20)q^{6} + -(74876497582956721159 + O(11^20))*a^2 + (128308652523796125897 + O(11^20))*a - 216030589636627076766 + O(11^20)q^{7} + (179872375456932504195 + O(11^20))*a^2 + (41167414242319362385 + O(11^20))*a - 210841057517064670475 + O(11^20)q^{8} + -(196227769989181940335 + O(11^20))*a^2 - (104526655010805199407 + O(11^20))*a + 197438649616604575264 + O(11^20)q^{9} + -(71391365500713487726 + O(11^20))*a^2 + (237140037334375000868 + O(11^20))*a + 136642671148158180752 + O(11^20)q^{10} + -(6471662588803686460*11 + O(11^20))*a^2 + (21245108018379929106*11 + O(11^20))*a - 20686351859699962235*11 + O(11^20)q^{11} + -(267173984533929290358 + O(11^20))*a^2 + (99754482923420751911 + O(11^20))*a + 141597952696203725694 + O(11^20)q^{12} + -815730721q^{13} + -(51096450356173876496 + O(11^20))*a^2 + (257257083675995671626 + O(11^20))*a + 237408682042918616888 + O(11^20)q^{14} + -(289934825349939680807 + O(11^20))*a^2 - (11000262763865428178*11 + O(11^20))*a + 224147636419749071020 + O(11^20)q^{15} + (295283955689600901776 + O(11^20))*a^2 - (20206118722874328192 + O(11^20))*a + 300346790331365632049 + O(11^20)q^{16} + (127107690642767188535 + O(11^20))*a^2 - (290403863786268544102 + O(11^20))*a - 206063594537721036723 + O(11^20)q^{17} + (35828085643784992027 + O(11^20))*a^2 - (217264071391869846631 + O(11^20))*a - 126469075039791961352 + O(11^20)q^{18} + (21240962067465993280 + O(11^20))*a^2 - (14742874762373252012 + O(11^20))*a + 113231826398361026205 + O(11^20)q^{19} + -(65135236818980544173 + O(11^20))*a^2 - (278827988407146949560 + O(11^20))*a - 7831435695833978885 + O(11^20)q^{20} + (97651087496701653768 + O(11^20))*a^2 - (5896933888367694243*11 + O(11^20))*a - 171489876588728548432 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -179872375456932504195 + O(11^20)x^{2} + -41167414242319624529 + O(11^20)x + 210841057517064670475 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7*z^5 + z^2 + 10*z + 4q^{2} + 10*z^5 + 7*z^4 + 7*z^3 + 2*z^2 + z + 4q^{3} + 3*z^4 + 3*z^3 + 9*z^2 + 8*z + 7q^{4} + 7*z^5 + 8*z^4 + 8*z^3 + 3*z^2 + 2*z + 5q^{5} + 4*z^5 + 8*z^4 + 8*z^3 + z^2 + 4*z + 10q^{6} + 4*z^5 + 5*z^4 + 5*z^3 + 3*z^2 + 7*z + 8q^{7} + 10*z^5 + 9*z^4 + 9*z^3 + 8*z^2 + 10*zq^{8} + 5*z^5 + z^4 + z^3 + 10*z^2 + 3*z + 4q^{9} + 8*z^5 + 5*z^4 + 5*z^3 + 2*z^2 + 8*z + 7q^{10} + 7*z^5 + 5*z^4 + 5*z^3 + 5*z^2 + 5*z + 3q^{12} + 8q^{13} + 9*z^5 + z^4 + z^3 + 9*z^2 + 4*z + 1q^{14} + 4*z^4 + 4*z^3 + z^2 + 7*z + 6q^{15} + z^5 + 4*z^4 + 4*z^3 + 9*z^2 + 10*zq^{16} + 7*z^5 + 4*z^4 + 4*z^3 + 2*z^2 + 6*z + 6q^{17} + 4*z^5 + 2*z^4 + 2*z^3 + 5*z^2 + 10*z + 2q^{18} + 4*z^5 + 8*z^4 + 8*z^3 + z^2 + 4*z + 7q^{19} + 3*z^5 + 8*z^4 + 8*z^3 + 4*z^2 + z + 8q^{20} + 10*z^4 + 10*z^3 + 8*z^2 + zq^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^97q^{2} + z^552q^{3} + z^426q^{4} + z^439q^{5} + z^649q^{6} + z^1121q^{7} + z^1238q^{8} + z^126q^{9} + z^536q^{10} + z^47q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1283)\omega^{6}&0\\0&u(z^47)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(131446899915469254352 + O(11^20))*a^2 - (16668102967084971608 + O(11^20))*a - 143028618626631968157 + O(11^20)q^{3} + a^2 - 131072q^{4} + (248431489477194627658 + O(11^20))*a^2 - (62985048310559850397 + O(11^20))*a - 107124721210528445277 + O(11^20)q^{5} + -(112777766827640977646 + O(11^20))*a^2 + (115594248386970322716 + O(11^20))*a - 65634966030451128723 + O(11^20)q^{6} + -(74876497582956721159 + O(11^20))*a^2 + (128308652523796125897 + O(11^20))*a - 216030589636627076766 + O(11^20)q^{7} + (179872375456932504195 + O(11^20))*a^2 + (41167414242319362385 + O(11^20))*a - 210841057517064670475 + O(11^20)q^{8} + -(196227769989181940335 + O(11^20))*a^2 - (104526655010805199407 + O(11^20))*a + 197438649616604575264 + O(11^20)q^{9} + -(71391365500713487726 + O(11^20))*a^2 + (237140037334375000868 + O(11^20))*a + 136642671148158180752 + O(11^20)q^{10} + -(6471662588803686460*11 + O(11^20))*a^2 + (21245108018379929106*11 + O(11^20))*a - 20686351859699962235*11 + O(11^20)q^{11} + -(267173984533929290358 + O(11^20))*a^2 + (99754482923420751911 + O(11^20))*a + 141597952696203725694 + O(11^20)q^{12} + -815730721q^{13} + -(51096450356173876496 + O(11^20))*a^2 + (257257083675995671626 + O(11^20))*a + 237408682042918616888 + O(11^20)q^{14} + -(289934825349939680807 + O(11^20))*a^2 - (11000262763865428178*11 + O(11^20))*a + 224147636419749071020 + O(11^20)q^{15} + (295283955689600901776 + O(11^20))*a^2 - (20206118722874328192 + O(11^20))*a + 300346790331365632049 + O(11^20)q^{16} + (127107690642767188535 + O(11^20))*a^2 - (290403863786268544102 + O(11^20))*a - 206063594537721036723 + O(11^20)q^{17} + (35828085643784992027 + O(11^20))*a^2 - (217264071391869846631 + O(11^20))*a - 126469075039791961352 + O(11^20)q^{18} + (21240962067465993280 + O(11^20))*a^2 - (14742874762373252012 + O(11^20))*a + 113231826398361026205 + O(11^20)q^{19} + -(65135236818980544173 + O(11^20))*a^2 - (278827988407146949560 + O(11^20))*a - 7831435695833978885 + O(11^20)q^{20} + (97651087496701653768 + O(11^20))*a^2 - (5896933888367694243*11 + O(11^20))*a - 171489876588728548432 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -179872375456932504195 + O(11^20)x^{2} + -41167414242319624529 + O(11^20)x + 210841057517064670475 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5*z^5 + 3*z^4 + 3*z^3 + 5*z^2 + z + 7q^{2} + 8*z^5 + 8*z^4 + 8*z^3 + 5*z + 6q^{3} + 8*z^5 + 9*z^4 + 9*z^3 + 3*z^2 + 4*z + 4q^{4} + 8*z^5 + 5*z^4 + 5*z^3 + 2*z^2 + 8*zq^{5} + 9*z^5 + 10*z^4 + 10*z^3 + 3*z^2 + 6*z + 10q^{6} + z^5 + z^4 + z^3 + 2*zq^{7} + 6*z^5 + 3*z^4 + 3*z^3 + 2*z^2 + 4*zq^{8} + z^5 + 2*z^4 + 2*z^3 + 3*z^2 + z + 2q^{9} + 7*z^5 + 9*z^4 + 9*z^3 + 6*z^2 + z + 7q^{10} + 7*z^4 + 7*z^3 + 10*z^2 + 4*z + 1q^{12} + 8q^{13} + 7*z^5 + 10*z^4 + 10*z^3 + 9*z^2 + 7q^{14} + 7*z^5 + z^4 + z^3 + 4*z^2 + 9*z + 2q^{15} + 3*z^5 + 3*z^4 + 3*z^3 + 6*z + 9q^{16} + z^5 + 4*z^4 + 4*z^3 + 9*z^2 + 10*z + 5q^{17} + 4*z^5 + 3*z^4 + 3*z^3 + 8*z^2 + 9*z + 8q^{18} + 9*z^5 + 10*z^4 + 10*z^3 + 3*z^2 + 6*z + 7q^{19} + 2*z^5 + 8*z^4 + 8*z^3 + 7*z^2 + 9*z + 6q^{20} + z^5 + 8*z^4 + 8*z^3 + 10*z^2 + 6*z + 1q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^1067q^{2} + z^752q^{3} + z^696q^{4} + z^839q^{5} + z^489q^{6} + z^361q^{7} + z^318q^{8} + z^56q^{9} + z^576q^{10} + z^517q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^813)\omega^{6}&0\\0&u(z^517)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(131446899915469254352 + O(11^20))*a^2 - (16668102967084971608 + O(11^20))*a - 143028618626631968157 + O(11^20)q^{3} + a^2 - 131072q^{4} + (248431489477194627658 + O(11^20))*a^2 - (62985048310559850397 + O(11^20))*a - 107124721210528445277 + O(11^20)q^{5} + -(112777766827640977646 + O(11^20))*a^2 + (115594248386970322716 + O(11^20))*a - 65634966030451128723 + O(11^20)q^{6} + -(74876497582956721159 + O(11^20))*a^2 + (128308652523796125897 + O(11^20))*a - 216030589636627076766 + O(11^20)q^{7} + (179872375456932504195 + O(11^20))*a^2 + (41167414242319362385 + O(11^20))*a - 210841057517064670475 + O(11^20)q^{8} + -(196227769989181940335 + O(11^20))*a^2 - (104526655010805199407 + O(11^20))*a + 197438649616604575264 + O(11^20)q^{9} + -(71391365500713487726 + O(11^20))*a^2 + (237140037334375000868 + O(11^20))*a + 136642671148158180752 + O(11^20)q^{10} + -(6471662588803686460*11 + O(11^20))*a^2 + (21245108018379929106*11 + O(11^20))*a - 20686351859699962235*11 + O(11^20)q^{11} + -(267173984533929290358 + O(11^20))*a^2 + (99754482923420751911 + O(11^20))*a + 141597952696203725694 + O(11^20)q^{12} + -815730721q^{13} + -(51096450356173876496 + O(11^20))*a^2 + (257257083675995671626 + O(11^20))*a + 237408682042918616888 + O(11^20)q^{14} + -(289934825349939680807 + O(11^20))*a^2 - (11000262763865428178*11 + O(11^20))*a + 224147636419749071020 + O(11^20)q^{15} + (295283955689600901776 + O(11^20))*a^2 - (20206118722874328192 + O(11^20))*a + 300346790331365632049 + O(11^20)q^{16} + (127107690642767188535 + O(11^20))*a^2 - (290403863786268544102 + O(11^20))*a - 206063594537721036723 + O(11^20)q^{17} + (35828085643784992027 + O(11^20))*a^2 - (217264071391869846631 + O(11^20))*a - 126469075039791961352 + O(11^20)q^{18} + (21240962067465993280 + O(11^20))*a^2 - (14742874762373252012 + O(11^20))*a + 113231826398361026205 + O(11^20)q^{19} + -(65135236818980544173 + O(11^20))*a^2 - (278827988407146949560 + O(11^20))*a - 7831435695833978885 + O(11^20)q^{20} + (97651087496701653768 + O(11^20))*a^2 - (5896933888367694243*11 + O(11^20))*a - 171489876588728548432 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -179872375456932504195 + O(11^20)x^{2} + -41167414242319624529 + O(11^20)x + 210841057517064670475 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 6q^{2} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*zq^{3} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 1q^{4} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 4q^{5} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*z + 1q^{6} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 2q^{7} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 3q^{8} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*zq^{9} + 8q^{10} + 4*z^5 + 2*z^4 + 4*z^3 + 8*z^2 + z + 10q^{12} + 8q^{13} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 8q^{14} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 10q^{15} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*z + 5q^{16} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 4q^{17} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 4q^{18} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 7q^{19} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 9q^{20} + 7*z^5 + 9*z^4 + 7*z^3 + 3*z^2 + 10*z + 8q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^31q^{2} + z^25q^{3} + z^22q^{4} + z^5q^{5} + z^56q^{6} + z^38q^{7} + z^50q^{8} + z^73q^{9} + 8q^{10} + z^65q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^55)\omega^{1}&0\\0&u(z^65)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(251157685445580017414 + O(11^20))*a - 211522979086522098313 + O(11^20)q^{3} + (278917505612855520135 + O(11^20))*a + 265801768396367255290 + O(11^20)q^{4} + (95897020197850878544 + O(11^20))*a - 168834143089696130454 + O(11^20)q^{5} + (241304523491755823630 + O(11^20))*a - 8989424310386254763 + O(11^20)q^{6} + (229375404370051590890 + O(11^20))*a - 162731959496640124197 + O(11^20)q^{7} + -(239147309550726811805 + O(11^20))*a - 84360658941818716108 + O(11^20)q^{8} + (320725115062636248902 + O(11^20))*a - 101278434917921209384 + O(11^20)q^{9} + (722302050528306190*11^2 + O(11^20))*a - 271905961476176858878 + O(11^20)q^{10} + (27995605320306607628*11 + O(11^20))*a - 661298934354984443*11^2 + O(11^20)q^{11} + (274239627763664940261 + O(11^20))*a + 38301770559711586666 + O(11^20)q^{12} + -815730721q^{13} + (234084917845784294543 + O(11^20))*a + 266349477935623914141 + O(11^20)q^{14} + -(103657139201945751489 + O(11^20))*a + 278945581592450353146 + O(11^20)q^{15} + (71504967044455391505 + O(11^20))*a - 72254448793848104632 + O(11^20)q^{16} + (95827848895805211868 + O(11^20))*a + 84196480384593878959 + O(11^20)q^{17} + -(51759889030508192918 + O(11^20))*a - 329528320418896970043 + O(11^20)q^{18} + (205191804344267328854 + O(11^20))*a - 290907798149866244748 + O(11^20)q^{19} + -(28299446300072876936 + O(11^20))*a - 167803151868169340494 + O(11^20)q^{20} + -(34110964855482709133 + O(11^20))*a - 90759072901225433777 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -278917505612855520135 + O(11^20)x + -265801768396367386362 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 4q^{2} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 4q^{3} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 3q^{4} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 6q^{5} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 6q^{6} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*z + 8q^{7} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 1q^{8} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 5q^{9} + 8q^{10} + 7*z^5 + 9*z^4 + 7*z^3 + 3*z^2 + 10*zq^{12} + 8q^{13} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 6q^{14} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 1q^{15} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 10q^{16} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 2q^{17} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 2q^{18} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*z + 3q^{19} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 1q^{20} + 4*z^5 + 2*z^4 + 4*z^3 + 8*z^2 + z + 7q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^101q^{2} + z^35q^{3} + z^2q^{4} + z^55q^{5} + z^16q^{6} + z^58q^{7} + z^70q^{8} + z^83q^{9} + 8q^{10} + z^115q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^5)\omega^{1}&0\\0&u(z^115)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(251157685445580017414 + O(11^20))*a - 211522979086522098313 + O(11^20)q^{3} + (278917505612855520135 + O(11^20))*a + 265801768396367255290 + O(11^20)q^{4} + (95897020197850878544 + O(11^20))*a - 168834143089696130454 + O(11^20)q^{5} + (241304523491755823630 + O(11^20))*a - 8989424310386254763 + O(11^20)q^{6} + (229375404370051590890 + O(11^20))*a - 162731959496640124197 + O(11^20)q^{7} + -(239147309550726811805 + O(11^20))*a - 84360658941818716108 + O(11^20)q^{8} + (320725115062636248902 + O(11^20))*a - 101278434917921209384 + O(11^20)q^{9} + (722302050528306190*11^2 + O(11^20))*a - 271905961476176858878 + O(11^20)q^{10} + (27995605320306607628*11 + O(11^20))*a - 661298934354984443*11^2 + O(11^20)q^{11} + (274239627763664940261 + O(11^20))*a + 38301770559711586666 + O(11^20)q^{12} + -815730721q^{13} + (234084917845784294543 + O(11^20))*a + 266349477935623914141 + O(11^20)q^{14} + -(103657139201945751489 + O(11^20))*a + 278945581592450353146 + O(11^20)q^{15} + (71504967044455391505 + O(11^20))*a - 72254448793848104632 + O(11^20)q^{16} + (95827848895805211868 + O(11^20))*a + 84196480384593878959 + O(11^20)q^{17} + -(51759889030508192918 + O(11^20))*a - 329528320418896970043 + O(11^20)q^{18} + (205191804344267328854 + O(11^20))*a - 290907798149866244748 + O(11^20)q^{19} + -(28299446300072876936 + O(11^20))*a - 167803151868169340494 + O(11^20)q^{20} + -(34110964855482709133 + O(11^20))*a - 90759072901225433777 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -278917505612855520135 + O(11^20)x + -265801768396367386362 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6*z^5 + 8*z^4 + 8*z^3 + 6*z^2 + 10*z + 4q^{2} + 8*z^5 + 8*z^4 + 8*z^3 + 5*z + 6q^{3} + 8*z^5 + 9*z^4 + 9*z^3 + 3*z^2 + 4*z + 4q^{4} + 8*z^5 + 5*z^4 + 5*z^3 + 2*z^2 + 8*zq^{5} + 2*z^5 + z^4 + z^3 + 8*z^2 + 5*z + 1q^{6} + 10*z^5 + 10*z^4 + 10*z^3 + 9*zq^{7} + 5*z^5 + 8*z^4 + 8*z^3 + 9*z^2 + 7*zq^{8} + z^5 + 2*z^4 + 2*z^3 + 3*z^2 + z + 2q^{9} + 4*z^5 + 2*z^4 + 2*z^3 + 5*z^2 + 10*z + 4q^{10} + 7*z^4 + 7*z^3 + 10*z^2 + 4*z + 1q^{12} + 3q^{13} + 7*z^5 + 10*z^4 + 10*z^3 + 9*z^2 + 7q^{14} + 7*z^5 + z^4 + z^3 + 4*z^2 + 9*z + 2q^{15} + 3*z^5 + 3*z^4 + 3*z^3 + 6*z + 9q^{16} + 10*z^5 + 7*z^4 + 7*z^3 + 2*z^2 + z + 6q^{17} + 7*z^5 + 8*z^4 + 8*z^3 + 3*z^2 + 2*z + 3q^{18} + 2*z^5 + z^4 + z^3 + 8*z^2 + 5*z + 4q^{19} + 2*z^5 + 8*z^4 + 8*z^3 + 7*z^2 + 9*z + 6q^{20} + 10*z^5 + 3*z^4 + 3*z^3 + z^2 + 5*z + 10q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^1067q^{2} + z^752q^{3} + z^696q^{4} + z^839q^{5} + z^489q^{6} + z^361q^{7} + z^318q^{8} + z^56q^{9} + z^576q^{10} + z^517q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^813)\omega^{1}&0\\0&u(z^517)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (149569575741345744067 + O(11^20))*a^2 + (55312490144383855402 + O(11^20))*a - 311262222554677784262 + O(11^20)q^{3} + a^2 - 131072q^{4} + (18735205580472032079 + O(11^20))*a^2 + (74535500139276212860 + O(11^20))*a - 112747231876766915259 + O(11^20)q^{5} + (301599169477698661086 + O(11^20))*a^2 - (250247632654785028979 + O(11^20))*a - 169236403449841771533 + O(11^20)q^{6} + -(14335411719530183775 + O(11^20))*a^2 - (71105348579061835897 + O(11^20))*a - 272972490743352732108 + O(11^20)q^{7} + (70877947609790528435 + O(11^20))*a^2 + (177822203473271662231 + O(11^20))*a - 308645731076688667003 + O(11^20)q^{8} + -(322659750603526840027 + O(11^20))*a^2 + (8969640084107277018 + O(11^20))*a - 259213134836902462758 + O(11^20)q^{9} + (190796495195215443151 + O(11^20))*a^2 - (198911597147031627410 + O(11^20))*a - 56043492265978822175 + O(11^20)q^{10} + -(26650494788326434517*11 + O(11^20))*a^2 - (25072698844281164140*11 + O(11^20))*a + 17376098921086446040*11 + O(11^20)q^{11} + -(244554450967432551860 + O(11^20))*a^2 + (311914471829335852487 + O(11^20))*a - 72058598017010758280 + O(11^20)q^{12} + 815730721q^{13} + (330330107196127219636 + O(11^20))*a^2 + (332321082741740077111 + O(11^20))*a + 20980894756417820882 + O(11^20)q^{14} + (218956033555645700275 + O(11^20))*a^2 - (2211839037262125497*11 + O(11^20))*a - 59683733851883073711 + O(11^20)q^{15} + (195343373811952570539 + O(11^20))*a^2 - (81941124735669415598 + O(11^20))*a + 66923247509356939137 + O(11^20)q^{16} + -(71777889533206699969 + O(11^20))*a^2 - (66884438390569181799 + O(11^20))*a - 148847017660804663957 + O(11^20)q^{17} + -(127958027037448893128 + O(11^20))*a^2 - (116326071986922203044 + O(11^20))*a + 94634718590851472095 + O(11^20)q^{18} + (11592447775930661168 + O(11^20))*a^2 + (262026825942027521220 + O(11^20))*a - 284814153681123821615 + O(11^20)q^{19} + -(111452875871509955905 + O(11^20))*a^2 - (88876602765122545672 + O(11^20))*a + 279884756033401996634 + O(11^20)q^{20} + (2398996655199154311 + O(11^20))*a^2 + (17729132352949607553*11 + O(11^20))*a - 261594364831706829605 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -70877947609790528435 + O(11^20)x^{2} + -177822203473271924375 + O(11^20)x + 308645731076688667003 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^5 + 3*z^4 + 3*z^3 + 6*z^2 + 8q^{2} + 4*z^5 + 7*z^4 + 7*z^3 + 9*z^2 + 5*z + 3q^{3} + 3*z^5 + 10*z^4 + 10*z^3 + 10*z^2 + 10*zq^{4} + 7*z^5 + 9*z^4 + 9*z^3 + 6*z^2 + zq^{5} + 2*z^5 + 7*z^4 + 7*z^3 + 4*z^2 + 10*z + 4q^{6} + 5*z^5 + 6*z^4 + 6*z^3 + 3*z^2 + 9*z + 10q^{7} + 5*z^5 + z^4 + z^3 + 10*z^2 + 3*z + 2q^{8} + 5*z^5 + 8*z^4 + 8*z^3 + 9*z^2 + 7*z + 2q^{9} + 4*z^5 + 3*z^4 + 3*z^3 + 8*z^2 + 9*z + 10q^{10} + 4*z^5 + 10*z^4 + 10*z^3 + 7*z^2 + 2*z + 5q^{12} + 3q^{13} + 6*z^5 + 4*z^2 + 7*zq^{14} + 4*z^5 + 6*z^4 + 6*z^3 + 6*z^2 + 6*z + 4q^{15} + 7*z^5 + 4*z^4 + 4*z^3 + 2*z^2 + 6*z + 1q^{16} + 8*z^5 + 8*z^4 + 8*z^3 + 5*z + 8q^{17} + 8*z^5 + 5*z^4 + 5*z^3 + 2*z^2 + 8*z + 9q^{18} + 2*z^5 + 7*z^4 + 7*z^3 + 4*z^2 + 10*z + 7q^{19} + 6*z^5 + 6*z^4 + 6*z^3 + z + 2q^{20} + z^5 + 7*z^4 + 7*z^3 + 7*z^2 + 7*z + 5q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^1097q^{2} + z^292q^{3} + z^1006q^{4} + z^1249q^{5} + z^59q^{6} + z^1311q^{7} + z^838q^{8} + z^616q^{9} + z^1016q^{10} + z^367q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^963)\omega^{1}&0\\0&u(z^367)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (149569575741345744067 + O(11^20))*a^2 + (55312490144383855402 + O(11^20))*a - 311262222554677784262 + O(11^20)q^{3} + a^2 - 131072q^{4} + (18735205580472032079 + O(11^20))*a^2 + (74535500139276212860 + O(11^20))*a - 112747231876766915259 + O(11^20)q^{5} + (301599169477698661086 + O(11^20))*a^2 - (250247632654785028979 + O(11^20))*a - 169236403449841771533 + O(11^20)q^{6} + -(14335411719530183775 + O(11^20))*a^2 - (71105348579061835897 + O(11^20))*a - 272972490743352732108 + O(11^20)q^{7} + (70877947609790528435 + O(11^20))*a^2 + (177822203473271662231 + O(11^20))*a - 308645731076688667003 + O(11^20)q^{8} + -(322659750603526840027 + O(11^20))*a^2 + (8969640084107277018 + O(11^20))*a - 259213134836902462758 + O(11^20)q^{9} + (190796495195215443151 + O(11^20))*a^2 - (198911597147031627410 + O(11^20))*a - 56043492265978822175 + O(11^20)q^{10} + -(26650494788326434517*11 + O(11^20))*a^2 - (25072698844281164140*11 + O(11^20))*a + 17376098921086446040*11 + O(11^20)q^{11} + -(244554450967432551860 + O(11^20))*a^2 + (311914471829335852487 + O(11^20))*a - 72058598017010758280 + O(11^20)q^{12} + 815730721q^{13} + (330330107196127219636 + O(11^20))*a^2 + (332321082741740077111 + O(11^20))*a + 20980894756417820882 + O(11^20)q^{14} + (218956033555645700275 + O(11^20))*a^2 - (2211839037262125497*11 + O(11^20))*a - 59683733851883073711 + O(11^20)q^{15} + (195343373811952570539 + O(11^20))*a^2 - (81941124735669415598 + O(11^20))*a + 66923247509356939137 + O(11^20)q^{16} + -(71777889533206699969 + O(11^20))*a^2 - (66884438390569181799 + O(11^20))*a - 148847017660804663957 + O(11^20)q^{17} + -(127958027037448893128 + O(11^20))*a^2 - (116326071986922203044 + O(11^20))*a + 94634718590851472095 + O(11^20)q^{18} + (11592447775930661168 + O(11^20))*a^2 + (262026825942027521220 + O(11^20))*a - 284814153681123821615 + O(11^20)q^{19} + -(111452875871509955905 + O(11^20))*a^2 - (88876602765122545672 + O(11^20))*a + 279884756033401996634 + O(11^20)q^{20} + (2398996655199154311 + O(11^20))*a^2 + (17729132352949607553*11 + O(11^20))*a - 261594364831706829605 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -70877947609790528435 + O(11^20)x^{2} + -177822203473271924375 + O(11^20)x + 308645731076688667003 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4*z^5 + 10*z^2 + z + 7q^{2} + 10*z^5 + 7*z^4 + 7*z^3 + 2*z^2 + z + 4q^{3} + 3*z^4 + 3*z^3 + 9*z^2 + 8*z + 7q^{4} + 7*z^5 + 8*z^4 + 8*z^3 + 3*z^2 + 2*z + 5q^{5} + 7*z^5 + 3*z^4 + 3*z^3 + 10*z^2 + 7*z + 1q^{6} + 7*z^5 + 6*z^4 + 6*z^3 + 8*z^2 + 4*z + 3q^{7} + z^5 + 2*z^4 + 2*z^3 + 3*z^2 + zq^{8} + 5*z^5 + z^4 + z^3 + 10*z^2 + 3*z + 4q^{9} + 3*z^5 + 6*z^4 + 6*z^3 + 9*z^2 + 3*z + 4q^{10} + 7*z^5 + 5*z^4 + 5*z^3 + 5*z^2 + 5*z + 3q^{12} + 3q^{13} + 9*z^5 + z^4 + z^3 + 9*z^2 + 4*z + 1q^{14} + 4*z^4 + 4*z^3 + z^2 + 7*z + 6q^{15} + z^5 + 4*z^4 + 4*z^3 + 9*z^2 + 10*zq^{16} + 4*z^5 + 7*z^4 + 7*z^3 + 9*z^2 + 5*z + 5q^{17} + 7*z^5 + 9*z^4 + 9*z^3 + 6*z^2 + z + 9q^{18} + 7*z^5 + 3*z^4 + 3*z^3 + 10*z^2 + 7*z + 4q^{19} + 3*z^5 + 8*z^4 + 8*z^3 + 4*z^2 + z + 8q^{20} + z^4 + z^3 + 3*z^2 + 10*zq^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^97q^{2} + z^552q^{3} + z^426q^{4} + z^439q^{5} + z^649q^{6} + z^1121q^{7} + z^1238q^{8} + z^126q^{9} + z^536q^{10} + z^47q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1283)\omega^{1}&0\\0&u(z^47)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (149569575741345744067 + O(11^20))*a^2 + (55312490144383855402 + O(11^20))*a - 311262222554677784262 + O(11^20)q^{3} + a^2 - 131072q^{4} + (18735205580472032079 + O(11^20))*a^2 + (74535500139276212860 + O(11^20))*a - 112747231876766915259 + O(11^20)q^{5} + (301599169477698661086 + O(11^20))*a^2 - (250247632654785028979 + O(11^20))*a - 169236403449841771533 + O(11^20)q^{6} + -(14335411719530183775 + O(11^20))*a^2 - (71105348579061835897 + O(11^20))*a - 272972490743352732108 + O(11^20)q^{7} + (70877947609790528435 + O(11^20))*a^2 + (177822203473271662231 + O(11^20))*a - 308645731076688667003 + O(11^20)q^{8} + -(322659750603526840027 + O(11^20))*a^2 + (8969640084107277018 + O(11^20))*a - 259213134836902462758 + O(11^20)q^{9} + (190796495195215443151 + O(11^20))*a^2 - (198911597147031627410 + O(11^20))*a - 56043492265978822175 + O(11^20)q^{10} + -(26650494788326434517*11 + O(11^20))*a^2 - (25072698844281164140*11 + O(11^20))*a + 17376098921086446040*11 + O(11^20)q^{11} + -(244554450967432551860 + O(11^20))*a^2 + (311914471829335852487 + O(11^20))*a - 72058598017010758280 + O(11^20)q^{12} + 815730721q^{13} + (330330107196127219636 + O(11^20))*a^2 + (332321082741740077111 + O(11^20))*a + 20980894756417820882 + O(11^20)q^{14} + (218956033555645700275 + O(11^20))*a^2 - (2211839037262125497*11 + O(11^20))*a - 59683733851883073711 + O(11^20)q^{15} + (195343373811952570539 + O(11^20))*a^2 - (81941124735669415598 + O(11^20))*a + 66923247509356939137 + O(11^20)q^{16} + -(71777889533206699969 + O(11^20))*a^2 - (66884438390569181799 + O(11^20))*a - 148847017660804663957 + O(11^20)q^{17} + -(127958027037448893128 + O(11^20))*a^2 - (116326071986922203044 + O(11^20))*a + 94634718590851472095 + O(11^20)q^{18} + (11592447775930661168 + O(11^20))*a^2 + (262026825942027521220 + O(11^20))*a - 284814153681123821615 + O(11^20)q^{19} + -(111452875871509955905 + O(11^20))*a^2 - (88876602765122545672 + O(11^20))*a + 279884756033401996634 + O(11^20)q^{20} + (2398996655199154311 + O(11^20))*a^2 + (17729132352949607553*11 + O(11^20))*a - 261594364831706829605 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -70877947609790528435 + O(11^20)x^{2} + -177822203473271924375 + O(11^20)x + 308645731076688667003 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 7q^{2} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 4q^{3} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 3q^{4} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 6q^{5} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*z + 5q^{6} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 3q^{7} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 10q^{8} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 5q^{9} + 3q^{10} + 7*z^5 + 9*z^4 + 7*z^3 + 3*z^2 + 10*zq^{12} + 3q^{13} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 6q^{14} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 1q^{15} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 10q^{16} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 9q^{17} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 9q^{18} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 8q^{19} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 1q^{20} + 7*z^5 + 9*z^4 + 7*z^3 + 3*z^2 + 10*z + 4q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^101q^{2} + z^35q^{3} + z^2q^{4} + z^55q^{5} + z^16q^{6} + z^58q^{7} + z^70q^{8} + z^83q^{9} + 8q^{10} + z^115q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^5)\omega^{6}&0\\0&u(z^115)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (218301364107339573073 + O(11^20))*a - 317426206022229757661 + O(11^20)q^{3} + (197909813640634484302 + O(11^20))*a - 39775139136064597346 + O(11^20)q^{4} + -(55623529301721924641 + O(11^20))*a - 18544470360882193960 + O(11^20)q^{5} + -(82871056430802840285 + O(11^20))*a - 311362745582668031367 + O(11^20)q^{6} + (330509496135486050391 + O(11^20))*a - 5158415913546529731 + O(11^20)q^{7} + -(234999919310435269341 + O(11^20))*a - 29602841799772490653 + O(11^20)q^{8} + -(35778824788734492864 + O(11^20))*a - 196825649541021105678 + O(11^20)q^{9} + -(15800652036949667864*11 + O(11^20))*a - 287147621133651957940 + O(11^20)q^{10} + (22783843855514819323*11 + O(11^20))*a - 12062036429396396128*11 + O(11^20)q^{11} + (130606226486869935027 + O(11^20))*a - 265516657881613764415 + O(11^20)q^{12} + 815730721q^{13} + -(51685024526844878038 + O(11^20))*a + 326796038033106269762 + O(11^20)q^{14} + (172902938990627226437 + O(11^20))*a + 221788832920340810184 + O(11^20)q^{15} + -(74775165204249792274 + O(11^20))*a + 42567000684170454534 + O(11^20)q^{16} + (234940862872644808364 + O(11^20))*a + 41759455222102692560 + O(11^20)q^{17} + -(310305400614851292650 + O(11^20))*a + 293919710981770294357 + O(11^20)q^{18} + (25287435991802393279 + O(11^20))*a - 321281036701597434380 + O(11^20)q^{19} + (121186396666003681371 + O(11^20))*a + 56889422011153215084 + O(11^20)q^{20} + (1800328397102481032 + O(11^20))*a - 199963198877026977364 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -197909813640634484302 + O(11^20)x + 39775139136064466274 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 5q^{2} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*zq^{3} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 1q^{4} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 4q^{5} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 10q^{6} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*z + 9q^{7} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 8q^{8} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*zq^{9} + 3q^{10} + 4*z^5 + 2*z^4 + 4*z^3 + 8*z^2 + z + 10q^{12} + 3q^{13} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 8q^{14} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 10q^{15} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*z + 5q^{16} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 7q^{17} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 7q^{18} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*z + 4q^{19} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 9q^{20} + 4*z^5 + 2*z^4 + 4*z^3 + 8*z^2 + z + 3q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^31q^{2} + z^25q^{3} + z^22q^{4} + z^5q^{5} + z^56q^{6} + z^38q^{7} + z^50q^{8} + z^73q^{9} + 8q^{10} + z^65q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^55)\omega^{6}&0\\0&u(z^65)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (218301364107339573073 + O(11^20))*a - 317426206022229757661 + O(11^20)q^{3} + (197909813640634484302 + O(11^20))*a - 39775139136064597346 + O(11^20)q^{4} + -(55623529301721924641 + O(11^20))*a - 18544470360882193960 + O(11^20)q^{5} + -(82871056430802840285 + O(11^20))*a - 311362745582668031367 + O(11^20)q^{6} + (330509496135486050391 + O(11^20))*a - 5158415913546529731 + O(11^20)q^{7} + -(234999919310435269341 + O(11^20))*a - 29602841799772490653 + O(11^20)q^{8} + -(35778824788734492864 + O(11^20))*a - 196825649541021105678 + O(11^20)q^{9} + -(15800652036949667864*11 + O(11^20))*a - 287147621133651957940 + O(11^20)q^{10} + (22783843855514819323*11 + O(11^20))*a - 12062036429396396128*11 + O(11^20)q^{11} + (130606226486869935027 + O(11^20))*a - 265516657881613764415 + O(11^20)q^{12} + 815730721q^{13} + -(51685024526844878038 + O(11^20))*a + 326796038033106269762 + O(11^20)q^{14} + (172902938990627226437 + O(11^20))*a + 221788832920340810184 + O(11^20)q^{15} + -(74775165204249792274 + O(11^20))*a + 42567000684170454534 + O(11^20)q^{16} + (234940862872644808364 + O(11^20))*a + 41759455222102692560 + O(11^20)q^{17} + -(310305400614851292650 + O(11^20))*a + 293919710981770294357 + O(11^20)q^{18} + (25287435991802393279 + O(11^20))*a - 321281036701597434380 + O(11^20)q^{19} + (121186396666003681371 + O(11^20))*a + 56889422011153215084 + O(11^20)q^{20} + (1800328397102481032 + O(11^20))*a - 199963198877026977364 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -197909813640634484302 + O(11^20)x + 39775139136064466274 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 9q^{3} + 10q^{4} + 5q^{5} + 8q^{6} + 4q^{7} + 6q^{8} + 2q^{10} + 2q^{12} + 6q^{13} + 6q^{14} + q^{15} + 4q^{16} + 8q^{19} + 6q^{20} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 6q^{2} + 4q^{3} + 6q^{4} + 4q^{5} + 2q^{6} + 9q^{7} + 10q^{8} + 2q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 116423195320276415688 + O(11^20)q^{2} + -224675068198352242176 + O(11^20)q^{3} + -150898831612306126702 + O(11^20)q^{4} + 91653315089999987021 + O(11^20)q^{5} + -255680284324054468739 + O(11^20)q^{6} + -180001910866333393284 + O(11^20)q^{7} + 26354215159703407466 + O(11^20)q^{8} + -23049344351147302029*11 + O(11^20)q^{9} + -333779947711910610557 + O(11^20)q^{10} + 7930710172591148955*11 + O(11^20)q^{11} + -96869971134591424966 + O(11^20)q^{12} + 10604499373q^{13} + 77611858350341397154 + O(11^20)q^{14} + 117263288228950941034 + O(11^20)q^{15} + -288003090393573503109 + O(11^20)q^{16} + 1563387279478799428*11^2 + O(11^20)q^{17} + 2874233011214049542*11 + O(11^20)q^{18} + 183201729836378404170 + O(11^20)q^{19} + -306889067529710389069 + O(11^20)q^{20} + -212987487555839193865 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^135q^{2} + z^972q^{3} + z^1048q^{4} + z^587q^{5} + z^1107q^{6} + z^147q^{7} + z^575q^{8} + z^1227q^{9} + z^722q^{10} + z^690q^{12} + 6q^{13} + z^282q^{14} + z^229q^{15} + z^1086q^{16} + z^189q^{17} + z^32q^{18} + z^974q^{19} + z^305q^{20} + z^1119q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^2q^{2} + z^1238q^{3} + z^782q^{4} + z^55q^{5} + z^1240q^{6} + z^546q^{7} + z^176q^{8} + z^429q^{9} + z^57q^{10} + z^66q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1264)\omega^{8}&0\\0&u(z^66)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (261944427659166551128 + O(11^20))*a^2 - (1972874748507731608*11^2 + O(11^20))*a + 49991356319423805930 + O(11^20)q^{3} + a^2 - 524288q^{4} + -(15629132323290452509 + O(11^20))*a^2 + (223651836423998699113 + O(11^20))*a - 332974324924084149294 + O(11^20)q^{5} + -(8951521684482309093 + O(11^20))*a^2 + (167099212046762761076 + O(11^20))*a - 30176124299100187435 + O(11^20)q^{6} + -(39771220601587384255 + O(11^20))*a^2 + (152277785942337391989 + O(11^20))*a - 9136556182243961791 + O(11^20)q^{7} + (211246717601930227913 + O(11^20))*a^2 - (24698822317382780936 + O(11^20))*a + 122246286436849476560 + O(11^20)q^{8} + (146836213696897186347 + O(11^20))*a^2 - (248191317888897727249 + O(11^20))*a + 32711340319554954509 + O(11^20)q^{9} + -(30126170303609906331*11 + O(11^20))*a^2 - (334779423914629599808 + O(11^20))*a + 231218683686685030648 + O(11^20)q^{10} + (27724926688272700048*11 + O(11^20))*a^2 - (22313158266492118952*11 + O(11^20))*a - 2085444958414109814*11^2 + O(11^20)q^{11} + (158513676505459650987 + O(11^20))*a^2 + (265122853337554474545 + O(11^20))*a + 262840387354599694039 + O(11^20)q^{12} + 10604499373q^{13} + (21892042919591779545*11 + O(11^20))*a^2 + (294884136818829750381 + O(11^20))*a + 220609876938828292459 + O(11^20)q^{14} + (128470773818748458112 + O(11^20))*a^2 + (165105518390895303192 + O(11^20))*a + 166545980152304693617 + O(11^20)q^{15} + (73314554574014209497 + O(11^20))*a^2 + (307209712803010740916 + O(11^20))*a + 2487159702599744143*11^2 + O(11^20)q^{16} + (125628264243904985558 + O(11^20))*a^2 + (578602854718024654*11 + O(11^20))*a + 103547999653940705064 + O(11^20)q^{17} + -(80571504130870246645 + O(11^20))*a^2 + (18159551125799218623 + O(11^20))*a + 46099678710407259426 + O(11^20)q^{18} + (273130915323254570617 + O(11^20))*a^2 + (220892462696867813243 + O(11^20))*a + 97628645360743973743 + O(11^20)q^{19} + (42125126896567443114 + O(11^20))*a^2 + (228488996609381764969 + O(11^20))*a + 237550397517689306781 + O(11^20)q^{20} + (224538403870890159547 + O(11^20))*a^2 - (153925696639809943195 + O(11^20))*a - 303652499311060775277 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -211246717601930227913 + O(11^20)x^{2} + 24698822317381732360 + O(11^20)x + -122246286436849476560 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^155q^{2} + z^52q^{3} + z^888q^{4} + z^1137q^{5} + z^207q^{6} + z^287q^{7} + z^1005q^{8} + z^197q^{9} + z^1292q^{10} + z^940q^{12} + 6q^{13} + z^442q^{14} + z^1189q^{15} + z^1306q^{16} + z^749q^{17} + z^352q^{18} + z^74q^{19} + z^695q^{20} + z^339q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^22q^{2} + z^318q^{3} + z^622q^{4} + z^605q^{5} + z^340q^{6} + z^686q^{7} + z^606q^{8} + z^729q^{9} + z^627q^{10} + z^726q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^604)\omega^{8}&0\\0&u(z^726)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (261944427659166551128 + O(11^20))*a^2 - (1972874748507731608*11^2 + O(11^20))*a + 49991356319423805930 + O(11^20)q^{3} + a^2 - 524288q^{4} + -(15629132323290452509 + O(11^20))*a^2 + (223651836423998699113 + O(11^20))*a - 332974324924084149294 + O(11^20)q^{5} + -(8951521684482309093 + O(11^20))*a^2 + (167099212046762761076 + O(11^20))*a - 30176124299100187435 + O(11^20)q^{6} + -(39771220601587384255 + O(11^20))*a^2 + (152277785942337391989 + O(11^20))*a - 9136556182243961791 + O(11^20)q^{7} + (211246717601930227913 + O(11^20))*a^2 - (24698822317382780936 + O(11^20))*a + 122246286436849476560 + O(11^20)q^{8} + (146836213696897186347 + O(11^20))*a^2 - (248191317888897727249 + O(11^20))*a + 32711340319554954509 + O(11^20)q^{9} + -(30126170303609906331*11 + O(11^20))*a^2 - (334779423914629599808 + O(11^20))*a + 231218683686685030648 + O(11^20)q^{10} + (27724926688272700048*11 + O(11^20))*a^2 - (22313158266492118952*11 + O(11^20))*a - 2085444958414109814*11^2 + O(11^20)q^{11} + (158513676505459650987 + O(11^20))*a^2 + (265122853337554474545 + O(11^20))*a + 262840387354599694039 + O(11^20)q^{12} + 10604499373q^{13} + (21892042919591779545*11 + O(11^20))*a^2 + (294884136818829750381 + O(11^20))*a + 220609876938828292459 + O(11^20)q^{14} + (128470773818748458112 + O(11^20))*a^2 + (165105518390895303192 + O(11^20))*a + 166545980152304693617 + O(11^20)q^{15} + (73314554574014209497 + O(11^20))*a^2 + (307209712803010740916 + O(11^20))*a + 2487159702599744143*11^2 + O(11^20)q^{16} + (125628264243904985558 + O(11^20))*a^2 + (578602854718024654*11 + O(11^20))*a + 103547999653940705064 + O(11^20)q^{17} + -(80571504130870246645 + O(11^20))*a^2 + (18159551125799218623 + O(11^20))*a + 46099678710407259426 + O(11^20)q^{18} + (273130915323254570617 + O(11^20))*a^2 + (220892462696867813243 + O(11^20))*a + 97628645360743973743 + O(11^20)q^{19} + (42125126896567443114 + O(11^20))*a^2 + (228488996609381764969 + O(11^20))*a + 237550397517689306781 + O(11^20)q^{20} + (224538403870890159547 + O(11^20))*a^2 - (153925696639809943195 + O(11^20))*a - 303652499311060775277 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -211246717601930227913 + O(11^20)x^{2} + 24698822317381732360 + O(11^20)x + -122246286436849476560 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^375q^{2} + z^572q^{3} + z^458q^{4} + z^537q^{5} + z^947q^{6} + z^497q^{7} + z^415q^{8} + z^837q^{9} + z^912q^{10} + z^1030q^{12} + 6q^{13} + z^872q^{14} + z^1109q^{15} + z^1066q^{16} + z^259q^{17} + z^1212q^{18} + z^814q^{19} + z^995q^{20} + z^1069q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^242q^{2} + z^838q^{3} + z^192q^{4} + z^5q^{5} + z^1080q^{6} + z^896q^{7} + z^16q^{8} + z^39q^{9} + z^247q^{10} + z^6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1324)\omega^{8}&0\\0&u(z^6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (261944427659166551128 + O(11^20))*a^2 - (1972874748507731608*11^2 + O(11^20))*a + 49991356319423805930 + O(11^20)q^{3} + a^2 - 524288q^{4} + -(15629132323290452509 + O(11^20))*a^2 + (223651836423998699113 + O(11^20))*a - 332974324924084149294 + O(11^20)q^{5} + -(8951521684482309093 + O(11^20))*a^2 + (167099212046762761076 + O(11^20))*a - 30176124299100187435 + O(11^20)q^{6} + -(39771220601587384255 + O(11^20))*a^2 + (152277785942337391989 + O(11^20))*a - 9136556182243961791 + O(11^20)q^{7} + (211246717601930227913 + O(11^20))*a^2 - (24698822317382780936 + O(11^20))*a + 122246286436849476560 + O(11^20)q^{8} + (146836213696897186347 + O(11^20))*a^2 - (248191317888897727249 + O(11^20))*a + 32711340319554954509 + O(11^20)q^{9} + -(30126170303609906331*11 + O(11^20))*a^2 - (334779423914629599808 + O(11^20))*a + 231218683686685030648 + O(11^20)q^{10} + (27724926688272700048*11 + O(11^20))*a^2 - (22313158266492118952*11 + O(11^20))*a - 2085444958414109814*11^2 + O(11^20)q^{11} + (158513676505459650987 + O(11^20))*a^2 + (265122853337554474545 + O(11^20))*a + 262840387354599694039 + O(11^20)q^{12} + 10604499373q^{13} + (21892042919591779545*11 + O(11^20))*a^2 + (294884136818829750381 + O(11^20))*a + 220609876938828292459 + O(11^20)q^{14} + (128470773818748458112 + O(11^20))*a^2 + (165105518390895303192 + O(11^20))*a + 166545980152304693617 + O(11^20)q^{15} + (73314554574014209497 + O(11^20))*a^2 + (307209712803010740916 + O(11^20))*a + 2487159702599744143*11^2 + O(11^20)q^{16} + (125628264243904985558 + O(11^20))*a^2 + (578602854718024654*11 + O(11^20))*a + 103547999653940705064 + O(11^20)q^{17} + -(80571504130870246645 + O(11^20))*a^2 + (18159551125799218623 + O(11^20))*a + 46099678710407259426 + O(11^20)q^{18} + (273130915323254570617 + O(11^20))*a^2 + (220892462696867813243 + O(11^20))*a + 97628645360743973743 + O(11^20)q^{19} + (42125126896567443114 + O(11^20))*a^2 + (228488996609381764969 + O(11^20))*a + 237550397517689306781 + O(11^20)q^{20} + (224538403870890159547 + O(11^20))*a^2 - (153925696639809943195 + O(11^20))*a - 303652499311060775277 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + -211246717601930227913 + O(11^20)x^{2} + 24698822317381732360 + O(11^20)x + -122246286436849476560 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 7q^{3} + 8q^{4} + 4q^{5} + 2q^{6} + 5q^{7} + 10q^{8} + q^{9} + 9q^{10} + q^{12} + 6q^{13} + 3q^{14} + 6q^{15} + 2q^{16} + 8q^{17} + 5q^{18} + 9q^{19} + 10q^{20} + 2q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 6q^{3} + 2q^{4} + 3q^{5} + 4q^{6} + 7q^{7} + 4q^{8} + 5q^{9} + 2q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{8}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -147619578742164675999 + O(11^20)q^{2} + 144268645987620008312 + O(11^20)q^{3} + 119694842743539212751 + O(11^20)q^{4} + 80305151870029437135 + O(11^20)q^{5} + -281438782306002185852 + O(11^20)q^{6} + 115296854666829851108 + O(11^20)q^{7} + -84901567145413118 + O(11^20)q^{8} + 154682306545564341985 + O(11^20)q^{9} + -277264805125848683794 + O(11^20)q^{10} + 25363522588800060763*11 + O(11^20)q^{11} + -79813011207949546262 + O(11^20)q^{12} + 10604499373q^{13} + 96362691120407490958 + O(11^20)q^{14} + 80050292556712691923 + O(11^20)q^{15} + 204861993258385865272 + O(11^20)q^{16} + 295848783646984984259 + O(11^20)q^{17} + 301646133348570110688 + O(11^20)q^{18} + -62837385514577433795 + O(11^20)q^{19} + 85842204075083025646 + O(11^20)q^{20} + 16393959567367867531 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + q^{3} + 9q^{4} + 10q^{5} + 9q^{6} + 8q^{7} + 5q^{8} + 8q^{9} + 2q^{10} + 9q^{12} + 5q^{13} + 6q^{14} + 10q^{15} + 2q^{16} + 6q^{18} + 10q^{19} + 2q^{20} + 8q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 10q^{2} + 4q^{3} + 5q^{4} + 2q^{5} + 7q^{6} + 9q^{7} + 2q^{8} + 7q^{9} + 9q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{8}&0\\0&u(10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -234404086842689588464 + O(11^20)q^{2} + 581446077887876323 + O(11^20)q^{3} + -135839871186645041242 + O(11^20)q^{4} + -72527316704641511802 + O(11^20)q^{5} + 235833727071564421545 + O(11^20)q^{6} + 166676482220483040363 + O(11^20)q^{7} + 17531890579277680886 + O(11^20)q^{8} + 290016715091590569761 + O(11^20)q^{9} + -110356065082175692580 + O(11^20)q^{10} + 18596051557893655597*11 + O(11^20)q^{11} + -226321428247023473542 + O(11^20)q^{12} + -10604499373q^{13} + -323210050963168389454 + O(11^20)q^{14} + -67502827752870582337 + O(11^20)q^{15} + -280767717922816025867 + O(11^20)q^{16} + -3468969837238988993*11 + O(11^20)q^{17} + -273725586650761777504 + O(11^20)q^{18} + 283271459612099869415 + O(11^20)q^{19} + 69562629007199173067 + O(11^20)q^{20} + -222249496473888872761 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 2q^{2} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 6q^{3} + 3*z^9 + 9*z^7 + 9*z^6 + 7*z^5 + 6*z^2 + 6*z + 4q^{4} + 3q^{5} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 7q^{6} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 7q^{7} + 10q^{8} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 5q^{9} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 6q^{10} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*z + 5q^{12} + 5q^{13} + z^9 + 3*z^7 + 3*z^6 + 6*z^5 + 2*z^2 + 2*z + 1q^{14} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 7q^{15} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 7q^{16} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 1q^{17} + 3*z^9 + 9*z^7 + 9*z^6 + 7*z^5 + 6*z^2 + 6*z + 6q^{18} + 10*z^9 + 8*z^7 + 8*z^6 + 5*z^5 + 9*z^2 + 9*z + 8q^{19} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 1q^{20} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*z + 7q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^89q^{2} + z^20q^{3} + z^117q^{4} + 5q^{5} + z^109q^{6} + z^63q^{7} + 4q^{8} + z^118q^{9} + z^17q^{10} + z^70q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^50)\omega^{8}&0\\0&u(z^70)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(272303827025241883805 + O(11^20))*a - 215322758263783466853 + O(11^20)q^{3} + (263073086349674588224 + O(11^20))*a + 119157449566346766688 + O(11^20)q^{4} + -(11792744442634703783*11 + O(11^20))*a + 2132467115334317787 + O(11^20)q^{5} + (307370063729186535967 + O(11^20))*a - 11527209846235776497 + O(11^20)q^{6} + -(311112583582943152740 + O(11^20))*a - 297917013299758162449 + O(11^20)q^{7} + (6192675389814108818*11 + O(11^20))*a - 27963725221396248802 + O(11^20)q^{8} + -(99936423499898059302 + O(11^20))*a - 312458379130186459655 + O(11^20)q^{9} + -(296923324283038982515 + O(11^20))*a + 5803528070715235176*11 + O(11^20)q^{10} + (11981597508268817670*11 + O(11^20))*a - 14854164576330428026*11 + O(11^20)q^{11} + -(252721356561247014070 + O(11^20))*a - 18072158248157614059*11 + O(11^20)q^{12} + -10604499373q^{13} + (63439165617033570498 + O(11^20))*a + 195799181015717743593 + O(11^20)q^{14} + -(129923307831437396636 + O(11^20))*a + 108587408843261519964 + O(11^20)q^{15} + (24214110984069740785 + O(11^20))*a + 146559214190227232717 + O(11^20)q^{16} + (173923144566889851136 + O(11^20))*a - 3175073011623829013 + O(11^20)q^{17} + (234154222990670536593 + O(11^20))*a + 995513828048219595 + O(11^20)q^{18} + (295851502337360175853 + O(11^20))*a + 316794284444217877848 + O(11^20)q^{19} + (327803949062047361867 + O(11^20))*a - 230261960691287358726 + O(11^20)q^{20} + (111073886043987669657 + O(11^20))*a - 15225312207865725271 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -263073086349674588224 + O(11^20)x + -119157449566347290976 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*z + 10q^{2} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 3q^{3} + z^9 + 3*z^7 + 3*z^6 + 6*z^5 + 2*z^2 + 2*z + 5q^{4} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 9q^{5} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 2q^{6} + 9q^{7} + 9q^{8} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 2q^{9} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 8q^{10} + 10*z^9 + 8*z^7 + 8*z^6 + 5*z^5 + 9*z^2 + 9*z + 8q^{12} + 5q^{13} + 3*z^9 + 9*z^7 + 9*z^6 + 7*z^5 + 6*z^2 + 6*z + 2q^{14} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 8q^{15} + 8*z^9 + 2*z^7 + 2*z^6 + 4*z^5 + 5*z^2 + 5*z + 5q^{16} + z^9 + 3*z^7 + 3*z^6 + 6*z^5 + 2*z^2 + 2*z + 3q^{17} + 10*z^9 + 8*z^7 + 8*z^6 + 5*z^5 + 9*z^2 + 9*z + 2q^{18} + 3*z^9 + 9*z^7 + 9*z^6 + 7*z^5 + 6*z^2 + 6*z + 4q^{19} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 8q^{20} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 5q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^97q^{2} + z^70q^{3} + z^93q^{4} + z^56q^{5} + z^47q^{6} + q^{7} + 4q^{8} + z^5q^{9} + z^33q^{10} + z^76q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^44)\omega^{1}&0\\0&u(z^76)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (209867710879757042690 + O(11^20))*a + 202154414094505092526 + O(11^20)q^{3} + (22774454677973748929 + O(11^20))*a - 59336859581788100381 + O(11^20)q^{4} + (287336633575499315780 + O(11^20))*a - 83613365379999442545 + O(11^20)q^{5} + -(195058502989790474194 + O(11^20))*a - 107160137793774659224 + O(11^20)q^{6} + (2563543920251190144*11 + O(11^20))*a - 208680264182933080760 + O(11^20)q^{7} + (4021886618287982123*11 + O(11^20))*a - 207115167455718804339 + O(11^20)q^{8} + (4319860259984061734 + O(11^20))*a - 193493551960292943636 + O(11^20)q^{9} + -(324398354493518668378 + O(11^20))*a - 137005028109894811940 + O(11^20)q^{10} + -(23393544352390912576*11 + O(11^20))*a + 24475679125356274941*11 + O(11^20)q^{11} + (204319014151714204465 + O(11^20))*a - 236580570750573332245 + O(11^20)q^{12} + -10604499373q^{13} + (285691909951492828837 + O(11^20))*a + 833866046998830099*11 + O(11^20)q^{14} + -(5995847152348734660 + O(11^20))*a - 193241371615301142445 + O(11^20)q^{15} + -(71555344044507954827 + O(11^20))*a + 156353267723444211625 + O(11^20)q^{16} + -(74223820962819400540 + O(11^20))*a + 335559460822518978073 + O(11^20)q^{17} + (157295347233511199974 + O(11^20))*a + 7886004069475144073 + O(11^20)q^{18} + -(159895703726444612181 + O(11^20))*a + 156054462680065417426 + O(11^20)q^{19} + -(258533341790640675313 + O(11^20))*a + 73464025562232043786 + O(11^20)q^{20} + -(42408452093388325539 + O(11^20))*a + 57105879574712163540 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -22774454677973748929 + O(11^20)x + 59336859581787576093 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 4q^{2} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 8q^{3} + 8*z^9 + 2*z^7 + 2*z^6 + 4*z^5 + 5*z^2 + 5*z + 5q^{4} + 3q^{5} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 5q^{6} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 10q^{7} + 10q^{8} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*zq^{9} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*z + 1q^{10} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 10q^{12} + 5q^{13} + 10*z^9 + 8*z^7 + 8*z^6 + 5*z^5 + 9*z^2 + 9*z + 5q^{14} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*z + 2q^{15} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 10q^{16} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*z + 7q^{17} + 8*z^9 + 2*z^7 + 2*z^6 + 4*z^5 + 5*z^2 + 5*z + 7q^{18} + z^9 + 3*z^7 + 3*z^6 + 6*z^5 + 2*z^2 + 2*z + 4q^{19} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 4q^{20} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 1q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^19q^{2} + z^100q^{3} + z^87q^{4} + 5q^{5} + z^119q^{6} + z^93q^{7} + 4q^{8} + z^98q^{9} + z^67q^{10} + z^50q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^70)\omega^{8}&0\\0&u(z^50)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(272303827025241883805 + O(11^20))*a - 215322758263783466853 + O(11^20)q^{3} + (263073086349674588224 + O(11^20))*a + 119157449566346766688 + O(11^20)q^{4} + -(11792744442634703783*11 + O(11^20))*a + 2132467115334317787 + O(11^20)q^{5} + (307370063729186535967 + O(11^20))*a - 11527209846235776497 + O(11^20)q^{6} + -(311112583582943152740 + O(11^20))*a - 297917013299758162449 + O(11^20)q^{7} + (6192675389814108818*11 + O(11^20))*a - 27963725221396248802 + O(11^20)q^{8} + -(99936423499898059302 + O(11^20))*a - 312458379130186459655 + O(11^20)q^{9} + -(296923324283038982515 + O(11^20))*a + 5803528070715235176*11 + O(11^20)q^{10} + (11981597508268817670*11 + O(11^20))*a - 14854164576330428026*11 + O(11^20)q^{11} + -(252721356561247014070 + O(11^20))*a - 18072158248157614059*11 + O(11^20)q^{12} + -10604499373q^{13} + (63439165617033570498 + O(11^20))*a + 195799181015717743593 + O(11^20)q^{14} + -(129923307831437396636 + O(11^20))*a + 108587408843261519964 + O(11^20)q^{15} + (24214110984069740785 + O(11^20))*a + 146559214190227232717 + O(11^20)q^{16} + (173923144566889851136 + O(11^20))*a - 3175073011623829013 + O(11^20)q^{17} + (234154222990670536593 + O(11^20))*a + 995513828048219595 + O(11^20)q^{18} + (295851502337360175853 + O(11^20))*a + 316794284444217877848 + O(11^20)q^{19} + (327803949062047361867 + O(11^20))*a - 230261960691287358726 + O(11^20)q^{20} + (111073886043987669657 + O(11^20))*a - 15225312207865725271 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -263073086349674588224 + O(11^20)x + -119157449566347290976 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 4q^{2} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*zq^{3} + 10*z^9 + 8*z^7 + 8*z^6 + 5*z^5 + 9*z^2 + 9*z + 9q^{4} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 1q^{5} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 5q^{6} + 9q^{7} + 9q^{8} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 4q^{9} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 10q^{10} + z^9 + 3*z^7 + 3*z^6 + 6*z^5 + 2*z^2 + 2*z + 4q^{12} + 5q^{13} + 8*z^9 + 2*z^7 + 2*z^6 + 4*z^5 + 5*z^2 + 5*z + 3q^{14} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*z + 3q^{15} + 3*z^9 + 9*z^7 + 9*z^6 + 7*z^5 + 6*z^2 + 6*z + 4q^{16} + 10*z^9 + 8*z^7 + 8*z^6 + 5*z^5 + 9*z^2 + 9*z + 7q^{17} + z^9 + 3*z^7 + 3*z^6 + 6*z^5 + 2*z^2 + 2*z + 9q^{18} + 8*z^9 + 2*z^7 + 2*z^6 + 4*z^5 + 5*z^2 + 5*z + 5q^{19} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 5q^{20} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*zq^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + z^107q^{2} + z^50q^{3} + z^63q^{4} + z^16q^{5} + z^37q^{6} + q^{7} + 4q^{8} + z^55q^{9} + z^3q^{10} + z^116q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^4)\omega^{1}&0\\0&u(z^116)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (209867710879757042690 + O(11^20))*a + 202154414094505092526 + O(11^20)q^{3} + (22774454677973748929 + O(11^20))*a - 59336859581788100381 + O(11^20)q^{4} + (287336633575499315780 + O(11^20))*a - 83613365379999442545 + O(11^20)q^{5} + -(195058502989790474194 + O(11^20))*a - 107160137793774659224 + O(11^20)q^{6} + (2563543920251190144*11 + O(11^20))*a - 208680264182933080760 + O(11^20)q^{7} + (4021886618287982123*11 + O(11^20))*a - 207115167455718804339 + O(11^20)q^{8} + (4319860259984061734 + O(11^20))*a - 193493551960292943636 + O(11^20)q^{9} + -(324398354493518668378 + O(11^20))*a - 137005028109894811940 + O(11^20)q^{10} + -(23393544352390912576*11 + O(11^20))*a + 24475679125356274941*11 + O(11^20)q^{11} + (204319014151714204465 + O(11^20))*a - 236580570750573332245 + O(11^20)q^{12} + -10604499373q^{13} + (285691909951492828837 + O(11^20))*a + 833866046998830099*11 + O(11^20)q^{14} + -(5995847152348734660 + O(11^20))*a - 193241371615301142445 + O(11^20)q^{15} + -(71555344044507954827 + O(11^20))*a + 156353267723444211625 + O(11^20)q^{16} + -(74223820962819400540 + O(11^20))*a + 335559460822518978073 + O(11^20)q^{17} + (157295347233511199974 + O(11^20))*a + 7886004069475144073 + O(11^20)q^{18} + -(159895703726444612181 + O(11^20))*a + 156054462680065417426 + O(11^20)q^{19} + -(258533341790640675313 + O(11^20))*a + 73464025562232043786 + O(11^20)q^{20} + -(42408452093388325539 + O(11^20))*a + 57105879574712163540 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -22774454677973748929 + O(11^20)x + 59336859581787576093 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4*z^19 + 9*z^18 + 8*z^16 + 6*z^15 + 10*z^13 + 10*z^12 + z^9 + 2*z^8 + 10*z^7 + 7*z^6 + 8*z^5 + 9*z^4 + 6*z^2 + 10*z + 3q^{2} + 4*z^19 + 7*z^18 + 6*z^17 + 10*z^16 + 9*z^15 + 3*z^13 + 9*z^12 + 10*z^11 + 4*z^10 + 9*z^9 + 3*z^8 + 7*z^7 + 3*z^6 + 4*z^5 + 8*z^4 + 10*z^3 + 7*z^2 + 3*z + 9q^{3} + 6*z^19 + 5*z^18 + 5*z^17 + 5*z^16 + 10*z^15 + 9*z^14 + 5*z^13 + 5*z^12 + 9*z^11 + 10*z^10 + 4*z^8 + 6*z^7 + 9*z^6 + 6*z^5 + 2*z^4 + 3*z^3 + 6*z^2 + 5*z + 6q^{4} + 8*z^19 + 6*z^18 + 10*z^17 + 10*z^16 + 2*z^14 + 4*z^13 + 9*z^12 + 10*z^11 + 2*z^10 + 8*z^9 + 3*z^8 + 3*z^7 + 2*z^6 + 6*z^5 + z^4 + 9*z^3 + 6*z^2 + 8*z + 1q^{5} + 4*z^19 + 2*z^18 + 10*z^17 + 3*z^16 + 7*z^15 + 6*z^14 + 10*z^13 + 9*z^12 + 2*z^11 + 8*z^10 + 4*z^8 + 5*z^7 + 7*z^6 + 4*z^5 + 4*z^4 + 4*z^3 + z^2 + 5*z + 9q^{6} + 5*z^19 + 4*z^18 + 8*z^17 + 4*z^16 + 8*z^15 + 8*z^14 + z^13 + 9*z^12 + 6*z^11 + z^10 + 9*z^9 + 3*z^7 + z^6 + 7*z^5 + 7*z^4 + 5*z^3 + 3*z^2 + 9*z + 5q^{7} + 4*z^19 + 4*z^18 + 8*z^17 + 7*z^16 + 8*z^15 + 10*z^14 + 10*z^13 + z^12 + 9*z^11 + 9*z^10 + 5*z^9 + 3*z^8 + 7*z^7 + 2*z^6 + 4*z^5 + 9*z^4 + 3*z^3 + 9*z^2 + 7*z + 5q^{8} + 5*z^18 + 5*z^17 + 5*z^16 + 4*z^15 + 8*z^14 + 5*z^13 + 2*z^12 + 5*z^11 + 5*z^10 + 4*z^9 + 6*z^8 + 8*z^7 + 3*z^6 + 3*z^5 + 9*z^4 + 5*z^3 + 10*z^2 + 3*z + 4q^{9} + 10*z^19 + 4*z^18 + z^17 + 10*z^15 + 5*z^14 + 8*z^13 + 8*z^11 + 7*z^10 + 9*z^8 + 4*z^7 + 5*z^6 + z^5 + 3*z^4 + 3*z^3 + 8*z^2 + 10*z + 7q^{10} + 6*z^19 + 4*z^18 + z^17 + 7*z^16 + z^15 + 10*z^14 + 7*z^13 + 8*z^12 + 9*z^11 + 9*z^10 + 4*z^9 + 8*z^8 + 9*z^7 + 2*z^6 + 6*z^5 + 9*z^4 + 5*z^3 + 6*z^2 + 6*z + 9q^{12} + 10q^{13} + 10*z^19 + 8*z^18 + 3*z^17 + 8*z^16 + 4*z^15 + 7*z^14 + 9*z^13 + 4*z^12 + 9*z^11 + z^10 + 5*z^9 + 10*z^8 + z^7 + 3*z^6 + 8*z^5 + 5*z^3 + 5*z^2 + 6*z + 3q^{14} + 4*z^19 + 5*z^18 + 7*z^17 + 2*z^16 + 8*z^15 + 10*z^14 + 9*z^12 + 7*z^11 + 6*z^10 + 8*z^9 + 3*z^8 + 8*z^7 + 4*z^6 + 8*z^5 + z^4 + 3*z^2 + 7*z + 2q^{15} + z^19 + 10*z^18 + 8*z^16 + 7*z^15 + 7*z^14 + 6*z^13 + 3*z^11 + 3*z^10 + 9*z^9 + 9*z^7 + 7*z^6 + 2*z^5 + 8*z^4 + 7*z^3 + z^2q^{16} + 8*z^19 + z^18 + 4*z^17 + 3*z^15 + 3*z^14 + 4*z^13 + 7*z^12 + 9*z^11 + 5*z^10 + z^9 + 9*z^7 + 9*z^6 + 6*z^5 + 5*z^4 + 6*z^3 + 8*z^2 + 3*z + 9q^{17} + 8*z^19 + 3*z^18 + 10*z^16 + 5*z^15 + 6*z^14 + 7*z^13 + 3*z^12 + 2*z^11 + 8*z^10 + 2*z^9 + 7*z^8 + 6*z^7 + 9*z^6 + 6*z^5 + 5*z^4 + 10*z^3 + 8*z + 4q^{18} + 7*z^19 + 8*z^18 + 8*z^17 + z^16 + 3*z^15 + 5*z^14 + 9*z^13 + 9*z^11 + 3*z^10 + 7*z^9 + 3*z^8 + 9*z^7 + 2*z^6 + 9*z^5 + 6*z^4 + 5*z^3 + 5*z^2 + z + 3q^{19} + 8*z^19 + 6*z^18 + 3*z^17 + 7*z^16 + z^15 + 5*z^14 + 3*z^13 + 6*z^12 + 5*z^11 + 8*z^10 + 2*z^9 + 6*z^8 + 2*z^7 + 9*z^6 + 4*z^5 + 8*z^4 + 3*z^3 + 10*z^2 + 8*z + 8q^{20} + z^19 + 8*z^18 + 5*z^16 + 9*z^15 + 10*z^14 + 5*z^13 + 8*z^12 + 4*z^11 + 7*z^10 + 9*z^9 + 7*z^8 + 2*z^7 + 10*z^6 + 8*z^5 + 7*z^4 + z^3 + 6*z^2 + 5*z + 5q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^142133q^{2} + z^40440q^{3} + z^141494q^{4} + z^103104q^{5} + z^21523q^{6} + z^153127q^{7} + z^95697q^{8} + z^99297q^{9} + z^84187q^{10} + z^66312q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^94738)\omega^{0}&0\\0&u(z^66312)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(126620356400921773329 + O(11^20))*a^4 + (116124275029971191609 + O(11^20))*a^3 + (54200904031282999190 + O(11^20))*a^2 + (27192497000010582578 + O(11^20))*a - 48718083868206747624 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(141869525449007678538 + O(11^20))*a^4 + (151605097979372508 + O(11^20))*a^3 - (194258345799019839578 + O(11^20))*a^2 - (13567311038568361865 + O(11^20))*a - 192798036755544856348 + O(11^20)q^{5} + -(166098107277332652573 + O(11^20))*a^4 + (262077544250279532283 + O(11^20))*a^3 - (261813859077022280337 + O(11^20))*a^2 + (39649627902202339891 + O(11^20))*a + 28297258544846119286 + O(11^20)q^{6} + -(27523841579649731365 + O(11^20))*a^4 + (20989109331104192790*11 + O(11^20))*a^3 - (116279107353648712844 + O(11^20))*a^2 - (232160258876642163398 + O(11^20))*a + 8449694136397374108 + O(11^20)q^{7} + a^3 - 4194304*aq^{8} + -(206728757606538797473 + O(11^20))*a^4 + (151039960356708325344 + O(11^20))*a^3 - (143140254869584811747 + O(11^20))*a^2 + (158708028880781775585 + O(11^20))*a + 258252730417370027917 + O(11^20)q^{9} + (2347130664133950252 + O(11^20))*a^4 - (273666526199816968253 + O(11^20))*a^3 - (146133167218784058890 + O(11^20))*a^2 - (168630994190763393409 + O(11^20))*a - 77518454593766032615 + O(11^20)q^{10} + (16934259358485004225*11 + O(11^20))*a^4 + (21643924749675957590*11 + O(11^20))*a^3 - (18870650781680015859*11 + O(11^20))*a^2 + (843567987186155850*11^2 + O(11^20))*a - 11614714628517476181*11 + O(11^20)q^{11} + -(39441557776730614930 + O(11^20))*a^4 - (179456845973125848329 + O(11^20))*a^3 - (66925150743570368834 + O(11^20))*a^2 - (245765224208019750989 + O(11^20))*a + 186082390215892588565 + O(11^20)q^{12} + -137858491849q^{13} + -(8392291303721118361 + O(11^20))*a^4 + (94080764106716336602 + O(11^20))*a^3 + (52533650851688319072 + O(11^20))*a^2 + (118488488431490924845 + O(11^20))*a + 17266215589407957762 + O(11^20)q^{14} + (130120802818579978503 + O(11^20))*a^4 + (284867432704604956532 + O(11^20))*a^3 + (132137272138135842803 + O(11^20))*a^2 + (287599407903525171023 + O(11^20))*a - 66108796264582541456 + O(11^20)q^{15} + a^4 - 6291456*a^2 + 4398046511104q^{16} + (247263884779259668634 + O(11^20))*a^4 - (199320934044999770198 + O(11^20))*a^3 + (335640184363866694800 + O(11^20))*a^2 + (111413043909931950062 + O(11^20))*a - 125895449116754295527 + O(11^20)q^{17} + -(129685198089365853755 + O(11^20))*a^4 + (61366251647099302067 + O(11^20))*a^3 - (145396434073872810176 + O(11^20))*a^2 + (289716777941210368284 + O(11^20))*a - 51076396781348042337 + O(11^20)q^{18} + -(102223264689557361929 + O(11^20))*a^4 + (332889877538916840034 + O(11^20))*a^3 + (155178524409514007361 + O(11^20))*a^2 + (236385904455101109948 + O(11^20))*a + 267406116352202105485 + O(11^20)q^{19} + -(106802364945428389007 + O(11^20))*a^4 - (73484333971793824548 + O(11^20))*a^3 + (248452910258062260177 + O(11^20))*a^2 - (237359369857924297976 + O(11^20))*a + 83016675446571761948 + O(11^20)q^{20} + (226558599902903013364 + O(11^20))*a^4 + (306754828431946001816 + O(11^20))*a^3 + (71714185102327232890 + O(11^20))*a^2 + (169717460484611971154 + O(11^20))*a + 26374001414417019471 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + 235939635302487581501 + O(11^20)x^{4} + 19832230278235438204 + O(11^20)x^{3} + -75305933702655388704 + O(11^20)x^{2} + -286001669361950811530 + O(11^20)x + -253175070989686225911 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5*z^19 + 8*z^18 + 2*z^17 + 5*z^16 + 3*z^15 + 5*z^14 + 9*z^13 + 2*z^12 + 6*z^11 + 4*z^10 + z^9 + 3*z^8 + 2*z^7 + 2*z^6 + 5*z^5 + 9*z^4 + z^3 + 8*z^2 + 6q^{2} + 4*z^19 + 8*z^18 + 3*z^17 + 5*z^16 + 8*z^15 + 2*z^14 + 3*z^12 + 5*z^11 + 10*z^9 + 2*z^8 + 3*z^7 + 6*z^6 + 2*z^5 + 8*z^4 + 2*z^3 + 2*z + 2q^{3} + 9*z^18 + 5*z^17 + 2*z^14 + 7*z^13 + 8*z^12 + 3*z^11 + 8*z^10 + 4*z^9 + 3*z^8 + 8*z^6 + 2*z^5 + 6*z^3 + 4*z + 6q^{4} + 9*z^19 + 6*z^18 + 5*z^16 + 9*z^15 + 2*z^13 + 7*z^12 + 3*z^11 + 10*z^10 + 10*z^9 + 3*z^8 + 7*z^7 + z^6 + 10*z^5 + 3*z^4 + z^3 + 9q^{5} + 7*z^19 + 3*z^18 + 9*z^17 + 6*z^16 + 9*z^15 + 8*z^14 + 2*z^13 + 4*z^12 + 2*z^11 + 6*z^10 + 2*z^9 + 4*z^8 + 5*z^7 + 10*z^6 + z^5 + 10*z^4 + 6*z^3 + 6*z^2 + 4*z + 2q^{6} + 4*z^19 + 7*z^18 + 3*z^17 + 3*z^16 + 7*z^15 + z^14 + 9*z^13 + 10*z^10 + z^9 + 2*z^8 + 5*z^7 + 9*z^6 + 10*z^5 + 4*z^4 + z^2 + 6*z + 4q^{7} + 3*z^19 + 8*z^18 + 5*z^17 + z^16 + 3*z^15 + 3*z^13 + 10*z^12 + 3*z^9 + z^8 + z^7 + 5*z^6 + 9*z^5 + 8*z^4 + 8*z^3 + 8*z^2 + 4q^{8} + 5*z^19 + 2*z^18 + 3*z^17 + 6*z^16 + 8*z^14 + 5*z^13 + 8*z^12 + 5*z^11 + 5*z^10 + 6*z^9 + z^8 + 8*z^6 + 3*z^5 + 6*z^4 + 4*z^3 + 4*z^2 + 2*z + 3q^{9} + 4*z^19 + 10*z^18 + 3*z^16 + z^15 + 10*z^14 + 2*z^13 + 3*z^12 + 5*z^11 + 3*z^10 + 5*z^9 + 8*z^8 + 8*z^7 + 3*z^6 + 4*z^5 + 8*z^4 + 5*z^3 + 6*z^2 + 5*z + 9q^{10} + 9*z^17 + 10*z^15 + 2*z^14 + 7*z^13 + 8*z^12 + 8*z^11 + 10*z^10 + 9*z^9 + 10*z^8 + 6*z^7 + z^6 + 5*z^5 + 8*z^4 + 6*z^3 + 10*z^2 + 10*z + 4q^{12} + 10q^{13} + 5*z^19 + 10*z^18 + 8*z^17 + 5*z^15 + 9*z^13 + z^12 + 3*z^11 + 10*z^10 + 6*z^9 + 9*z^8 + 2*z^7 + 9*z^6 + 9*z^5 + 7*z^4 + 5*z^3 + 4*z^2 + 5q^{14} + z^19 + 3*z^18 + 2*z^15 + 8*z^14 + 10*z^13 + 8*z^12 + z^11 + 10*z^10 + 10*z^9 + 9*z^8 + z^7 + 10*z^6 + 9*z^5 + 3*z^3 + 4*z^2 + 10*z + 3q^{15} + 7*z^19 + 10*z^18 + 5*z^17 + 5*z^16 + 9*z^15 + 10*z^14 + 7*z^13 + 7*z^12 + 9*z^11 + 9*z^10 + 7*z^8 + 3*z^7 + 3*z^6 + 6*z^5 + 4*z^4 + 7*z^3 + 7*z^2 + 8*z + 4q^{16} + 8*z^19 + 10*z^18 + 10*z^17 + z^16 + 2*z^15 + 9*z^14 + 5*z^13 + 3*z^12 + 8*z^11 + 3*z^10 + 2*z^9 + 5*z^8 + 6*z^7 + 8*z^6 + z^5 + 8*z^4 + 7*z^3 + 6*z^2 + 10*z + 2q^{17} + z^19 + 8*z^18 + 9*z^17 + 6*z^16 + z^15 + 6*z^14 + 4*z^13 + 8*z^12 + z^11 + z^10 + 3*z^9 + 2*z^8 + 8*z^7 + 8*z^6 + 3*z^5 + 5*z^3 + 8*z^2 + zq^{18} + 4*z^19 + 9*z^18 + 10*z^17 + 3*z^16 + 2*z^15 + 9*z^14 + 4*z^13 + 10*z^12 + 4*z^11 + 8*z^10 + 9*z^9 + 5*z^8 + 2*z^7 + 6*z^6 + 2*z^3 + 7*z^2 + 8*z + 10q^{19} + 4*z^19 + 10*z^18 + z^17 + 9*z^16 + 9*z^15 + 6*z^14 + 6*z^12 + z^11 + z^10 + 4*z^9 + z^8 + 5*z^7 + z^6 + 2*z^5 + 2*z^4 + z^3 + 8*z^2 + 4*z + 6q^{20} + 8*z^19 + 6*z^18 + 2*z^17 + 3*z^16 + z^15 + 8*z^14 + 9*z^12 + 9*z^11 + 6*z^9 + 5*z^8 + 5*z^7 + 5*z^6 + 8*z^5 + 6*z^4 + 3*z^3 + 3*z^2 + 3*z + 4q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^106323q^{2} + z^34940q^{3} + z^61064q^{4} + z^16824q^{5} + z^141263q^{6} + z^83787q^{7} + z^143207q^{8} + z^103307q^{9} + z^123147q^{10} + z^5872q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^155178)\omega^{0}&0\\0&u(z^5872)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(126620356400921773329 + O(11^20))*a^4 + (116124275029971191609 + O(11^20))*a^3 + (54200904031282999190 + O(11^20))*a^2 + (27192497000010582578 + O(11^20))*a - 48718083868206747624 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(141869525449007678538 + O(11^20))*a^4 + (151605097979372508 + O(11^20))*a^3 - (194258345799019839578 + O(11^20))*a^2 - (13567311038568361865 + O(11^20))*a - 192798036755544856348 + O(11^20)q^{5} + -(166098107277332652573 + O(11^20))*a^4 + (262077544250279532283 + O(11^20))*a^3 - (261813859077022280337 + O(11^20))*a^2 + (39649627902202339891 + O(11^20))*a + 28297258544846119286 + O(11^20)q^{6} + -(27523841579649731365 + O(11^20))*a^4 + (20989109331104192790*11 + O(11^20))*a^3 - (116279107353648712844 + O(11^20))*a^2 - (232160258876642163398 + O(11^20))*a + 8449694136397374108 + O(11^20)q^{7} + a^3 - 4194304*aq^{8} + -(206728757606538797473 + O(11^20))*a^4 + (151039960356708325344 + O(11^20))*a^3 - (143140254869584811747 + O(11^20))*a^2 + (158708028880781775585 + O(11^20))*a + 258252730417370027917 + O(11^20)q^{9} + (2347130664133950252 + O(11^20))*a^4 - (273666526199816968253 + O(11^20))*a^3 - (146133167218784058890 + O(11^20))*a^2 - (168630994190763393409 + O(11^20))*a - 77518454593766032615 + O(11^20)q^{10} + (16934259358485004225*11 + O(11^20))*a^4 + (21643924749675957590*11 + O(11^20))*a^3 - (18870650781680015859*11 + O(11^20))*a^2 + (843567987186155850*11^2 + O(11^20))*a - 11614714628517476181*11 + O(11^20)q^{11} + -(39441557776730614930 + O(11^20))*a^4 - (179456845973125848329 + O(11^20))*a^3 - (66925150743570368834 + O(11^20))*a^2 - (245765224208019750989 + O(11^20))*a + 186082390215892588565 + O(11^20)q^{12} + -137858491849q^{13} + -(8392291303721118361 + O(11^20))*a^4 + (94080764106716336602 + O(11^20))*a^3 + (52533650851688319072 + O(11^20))*a^2 + (118488488431490924845 + O(11^20))*a + 17266215589407957762 + O(11^20)q^{14} + (130120802818579978503 + O(11^20))*a^4 + (284867432704604956532 + O(11^20))*a^3 + (132137272138135842803 + O(11^20))*a^2 + (287599407903525171023 + O(11^20))*a - 66108796264582541456 + O(11^20)q^{15} + a^4 - 6291456*a^2 + 4398046511104q^{16} + (247263884779259668634 + O(11^20))*a^4 - (199320934044999770198 + O(11^20))*a^3 + (335640184363866694800 + O(11^20))*a^2 + (111413043909931950062 + O(11^20))*a - 125895449116754295527 + O(11^20)q^{17} + -(129685198089365853755 + O(11^20))*a^4 + (61366251647099302067 + O(11^20))*a^3 - (145396434073872810176 + O(11^20))*a^2 + (289716777941210368284 + O(11^20))*a - 51076396781348042337 + O(11^20)q^{18} + -(102223264689557361929 + O(11^20))*a^4 + (332889877538916840034 + O(11^20))*a^3 + (155178524409514007361 + O(11^20))*a^2 + (236385904455101109948 + O(11^20))*a + 267406116352202105485 + O(11^20)q^{19} + -(106802364945428389007 + O(11^20))*a^4 - (73484333971793824548 + O(11^20))*a^3 + (248452910258062260177 + O(11^20))*a^2 - (237359369857924297976 + O(11^20))*a + 83016675446571761948 + O(11^20)q^{20} + (226558599902903013364 + O(11^20))*a^4 + (306754828431946001816 + O(11^20))*a^3 + (71714185102327232890 + O(11^20))*a^2 + (169717460484611971154 + O(11^20))*a + 26374001414417019471 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + 235939635302487581501 + O(11^20)x^{4} + 19832230278235438204 + O(11^20)x^{3} + -75305933702655388704 + O(11^20)x^{2} + -286001669361950811530 + O(11^20)x + -253175070989686225911 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6*z^19 + 3*z^18 + 10*z^17 + 6*z^16 + 3*z^15 + 5*z^14 + 4*z^13 + 5*z^12 + z^11 + 2*z^10 + 3*z^8 + 6*z^7 + 3*z^6 + 9*z^5 + 9*z^4 + 7*z^3 + 9*z^2 + 6*z + 3q^{2} + 5*z^19 + 9*z^18 + 10*z^17 + 6*z^16 + 4*z^15 + 4*z^14 + 2*z^13 + 3*z^12 + z^11 + 3*z^10 + 9*z^7 + 9*z^6 + 9*z^5 + 7*z^4 + 3*z^3 + 8*z^2 + 3*z + 9q^{3} + 6*z^19 + 8*z^17 + 6*z^16 + z^15 + z^14 + z^13 + 5*z^11 + z^10 + 2*z^9 + 7*z^7 + 6*z^6 + 3*z^4 + 6*z^3 + 7*z^2 + 10*z + 10q^{4} + 8*z^19 + 9*z^18 + 10*z^17 + 3*z^16 + 6*z^15 + 6*z^14 + 4*z^13 + z^12 + 3*z^11 + 4*z^10 + 10*z^9 + 8*z^7 + 10*z^6 + 2*z^5 + 10*z^4 + 8*z^3 + 8*z^2 + 2*z + 1q^{5} + 5*z^19 + 3*z^18 + 8*z^16 + 5*z^15 + z^14 + z^13 + z^12 + 10*z^10 + z^9 + 5*z^8 + 7*z^7 + z^6 + 8*z^5 + 4*z^3 + 7*z^2 + 2*z + 3q^{6} + 4*z^19 + 3*z^18 + z^17 + z^16 + 4*z^15 + 7*z^14 + 4*z^13 + 3*z^12 + 10*z^11 + 8*z^10 + 5*z^9 + 10*z^8 + 8*z^7 + 4*z^6 + 9*z^5 + 5*z^4 + 8*z^3 + 3*z + 8q^{7} + z^19 + 9*z^18 + 8*z^16 + 3*z^15 + 5*z^14 + 10*z^13 + 3*z^12 + 9*z^11 + 3*z^10 + 3*z^9 + 3*z^8 + 4*z^6 + z^5 + 7*z^4 + z^3 + 8*z^2 + 9*z + 4q^{8} + 10*z^18 + 9*z^17 + 7*z^16 + z^15 + 2*z^14 + 10*z^13 + 10*z^12 + 3*z^11 + 8*z^10 + 8*z^9 + 7*z^8 + 8*z^7 + 10*z^6 + z^4 + 8*z^3 + 5*z^2 + 9*z + 8q^{9} + 7*z^19 + 2*z^18 + z^17 + 10*z^15 + 6*z^14 + 5*z^13 + 10*z^12 + 7*z^11 + 10*z^10 + z^9 + 5*z^8 + 9*z^7 + 7*z^6 + 3*z^5 + 10*z^4 + 3*z^3 + 2*z^2 + 5*z + 10q^{10} + 2*z^19 + 9*z^18 + 2*z^17 + 5*z^16 + 4*z^15 + z^14 + 5*z^13 + 3*z^12 + 9*z^11 + 7*z^10 + 9*z^9 + z^8 + z^7 + 2*z^6 + 10*z^5 + 8*z^4 + 8*z^3 + 3*z^2 + z + 10q^{12} + 10q^{13} + 3*z^19 + 8*z^18 + 10*z^17 + 7*z^16 + 2*z^15 + 3*z^14 + 9*z^13 + 8*z^12 + 2*z^11 + 6*z^9 + 7*z^8 + 8*z^7 + z^6 + 8*z^5 + 8*z^4 + 3*z^3 + z^2 + z + 8q^{14} + 8*z^18 + 7*z^17 + 8*z^15 + 9*z^14 + 4*z^13 + 3*z^12 + z^11 + 9*z^10 + 7*z^8 + 7*z^7 + 4*z^6 + 7*z^5 + 2*z^4 + 5*z^3 + 8*z^2 + 3*zq^{15} + 3*z^19 + z^18 + 3*z^17 + 8*z^16 + 2*z^15 + z^14 + 4*z^13 + 5*z^12 + 6*z^11 + 8*z^10 + 8*z^9 + 4*z^8 + 10*z^7 + 8*z^6 + 3*z^5 + 3*z^4 + 2*z^3 + 8*z + 2q^{16} + 7*z^19 + 5*z^18 + 10*z^17 + 3*z^16 + 7*z^14 + 9*z^13 + 7*z^12 + 8*z^11 + 5*z^10 + 5*z^9 + 5*z^8 + 9*z^7 + 8*z^6 + 3*z^5 + 9*z^4 + 10*z^3 + z + 2q^{17} + 5*z^19 + 7*z^18 + z^17 + 7*z^16 + z^15 + 3*z^14 + 6*z^13 + 4*z^12 + 5*z^11 + 10*z^10 + 8*z^9 + 3*z^8 + 7*z^7 + 10*z^6 + 3*z^5 + 4*z^4 + 5*z^3 + 4*z^2 + 8*z + 4q^{18} + 8*z^19 + 4*z^18 + 3*z^17 + 9*z^16 + z^15 + 6*z^13 + 7*z^12 + 6*z^11 + 9*z^10 + 4*z^9 + 4*z^8 + 6*z^7 + 8*z^6 + 4*z^5 + 9*z^4 + 9*z^3 + 8*z^2 + 9*z + 7q^{19} + 9*z^19 + 8*z^18 + 8*z^17 + 10*z^16 + 8*z^15 + 10*z^14 + 3*z^13 + 6*z^12 + 3*z^11 + 10*z^9 + 3*z^8 + z^7 + 10*z^6 + 8*z^5 + 8*z^4 + z^3 + 5*z^2 + 4*z + 3q^{20} + 5*z^19 + 7*z^18 + 10*z^17 + 5*z^15 + 3*z^14 + 3*z^13 + 2*z^12 + 2*z^11 + z^9 + 8*z^8 + 2*z^7 + 10*z^6 + z^5 + 7*z^4 + 3*z^3 + 4*z^2 + 6*z + 7q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^42203q^{2} + z^62240q^{3} + z^27504q^{4} + z^24014q^{5} + z^104443q^{6} + z^116407q^{7} + z^125827q^{8} + z^9027q^{9} + z^66217q^{10} + z^64592q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^96458)\omega^{0}&0\\0&u(z^64592)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(126620356400921773329 + O(11^20))*a^4 + (116124275029971191609 + O(11^20))*a^3 + (54200904031282999190 + O(11^20))*a^2 + (27192497000010582578 + O(11^20))*a - 48718083868206747624 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(141869525449007678538 + O(11^20))*a^4 + (151605097979372508 + O(11^20))*a^3 - (194258345799019839578 + O(11^20))*a^2 - (13567311038568361865 + O(11^20))*a - 192798036755544856348 + O(11^20)q^{5} + -(166098107277332652573 + O(11^20))*a^4 + (262077544250279532283 + O(11^20))*a^3 - (261813859077022280337 + O(11^20))*a^2 + (39649627902202339891 + O(11^20))*a + 28297258544846119286 + O(11^20)q^{6} + -(27523841579649731365 + O(11^20))*a^4 + (20989109331104192790*11 + O(11^20))*a^3 - (116279107353648712844 + O(11^20))*a^2 - (232160258876642163398 + O(11^20))*a + 8449694136397374108 + O(11^20)q^{7} + a^3 - 4194304*aq^{8} + -(206728757606538797473 + O(11^20))*a^4 + (151039960356708325344 + O(11^20))*a^3 - (143140254869584811747 + O(11^20))*a^2 + (158708028880781775585 + O(11^20))*a + 258252730417370027917 + O(11^20)q^{9} + (2347130664133950252 + O(11^20))*a^4 - (273666526199816968253 + O(11^20))*a^3 - (146133167218784058890 + O(11^20))*a^2 - (168630994190763393409 + O(11^20))*a - 77518454593766032615 + O(11^20)q^{10} + (16934259358485004225*11 + O(11^20))*a^4 + (21643924749675957590*11 + O(11^20))*a^3 - (18870650781680015859*11 + O(11^20))*a^2 + (843567987186155850*11^2 + O(11^20))*a - 11614714628517476181*11 + O(11^20)q^{11} + -(39441557776730614930 + O(11^20))*a^4 - (179456845973125848329 + O(11^20))*a^3 - (66925150743570368834 + O(11^20))*a^2 - (245765224208019750989 + O(11^20))*a + 186082390215892588565 + O(11^20)q^{12} + -137858491849q^{13} + -(8392291303721118361 + O(11^20))*a^4 + (94080764106716336602 + O(11^20))*a^3 + (52533650851688319072 + O(11^20))*a^2 + (118488488431490924845 + O(11^20))*a + 17266215589407957762 + O(11^20)q^{14} + (130120802818579978503 + O(11^20))*a^4 + (284867432704604956532 + O(11^20))*a^3 + (132137272138135842803 + O(11^20))*a^2 + (287599407903525171023 + O(11^20))*a - 66108796264582541456 + O(11^20)q^{15} + a^4 - 6291456*a^2 + 4398046511104q^{16} + (247263884779259668634 + O(11^20))*a^4 - (199320934044999770198 + O(11^20))*a^3 + (335640184363866694800 + O(11^20))*a^2 + (111413043909931950062 + O(11^20))*a - 125895449116754295527 + O(11^20)q^{17} + -(129685198089365853755 + O(11^20))*a^4 + (61366251647099302067 + O(11^20))*a^3 - (145396434073872810176 + O(11^20))*a^2 + (289716777941210368284 + O(11^20))*a - 51076396781348042337 + O(11^20)q^{18} + -(102223264689557361929 + O(11^20))*a^4 + (332889877538916840034 + O(11^20))*a^3 + (155178524409514007361 + O(11^20))*a^2 + (236385904455101109948 + O(11^20))*a + 267406116352202105485 + O(11^20)q^{19} + -(106802364945428389007 + O(11^20))*a^4 - (73484333971793824548 + O(11^20))*a^3 + (248452910258062260177 + O(11^20))*a^2 - (237359369857924297976 + O(11^20))*a + 83016675446571761948 + O(11^20)q^{20} + (226558599902903013364 + O(11^20))*a^4 + (306754828431946001816 + O(11^20))*a^3 + (71714185102327232890 + O(11^20))*a^2 + (169717460484611971154 + O(11^20))*a + 26374001414417019471 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + 235939635302487581501 + O(11^20)x^{4} + 19832230278235438204 + O(11^20)x^{3} + -75305933702655388704 + O(11^20)x^{2} + -286001669361950811530 + O(11^20)x + -253175070989686225911 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3*z^19 + 2*z^18 + 5*z^17 + 9*z^16 + 6*z^14 + 7*z^13 + 8*z^12 + 3*z^11 + 4*z^10 + z^9 + 9*z^8 + 2*z^7 + 7*z^6 + 3*z^4 + 8*z^2 + 8*z + 9q^{2} + 10*z^19 + 10*z^18 + 3*z^17 + 3*z^16 + 3*z^15 + 8*z^14 + 8*z^13 + 5*z^12 + 8*z^11 + 4*z^10 + 4*z^9 + 2*z^8 + 8*z^7 + 2*z^6 + 5*z^5 + 5*z^3 + 9*z^2 + 3*zq^{3} + 7*z^19 + 9*z^18 + 6*z^17 + z^16 + 6*z^15 + 9*z^14 + z^13 + 5*z^12 + 2*z^10 + 10*z^8 + 2*z^7 + 6*z^6 + 4*z^5 + 4*z^4 + 6*z^3 + 8*z^2 + 2*z + 10q^{4} + 2*z^19 + 4*z^18 + 9*z^17 + 3*z^16 + 7*z^15 + 7*z^14 + 6*z^12 + 4*z^10 + 5*z^9 + z^7 + 6*z^6 + 2*z^5 + 5*z^4 + 5*z^3 + 8*z^2 + 6*z + 7q^{5} + 9*z^19 + 4*z^17 + 3*z^14 + 8*z^13 + 5*z^12 + z^11 + 4*z^10 + 9*z^9 + 10*z^8 + 7*z^7 + 7*z^6 + z^4 + 7*z^3 + 7*z^2 + 5*z + 10q^{6} + 4*z^19 + 8*z^18 + 7*z^17 + z^15 + 6*z^14 + 4*z^13 + 5*z^12 + 5*z^10 + z^9 + 2*z^8 + 2*z^7 + 8*z^6 + 9*z^5 + z^4 + 9*z^2 + 8*z + 10q^{7} + 2*z^19 + 10*z^18 + z^17 + 8*z^16 + 3*z^15 + 8*z^14 + 8*z^13 + 2*z^12 + 7*z^11 + 8*z^10 + 2*z^9 + 2*z^8 + 2*z^7 + 2*z^6 + z^4 + 10*z^2 + 10*z + 1q^{8} + 3*z^19 + 9*z^18 + z^17 + z^16 + 5*z^15 + 6*z^14 + 2*z^13 + z^12 + 10*z^11 + 9*z^10 + 5*z^9 + 7*z^8 + 10*z^6 + 9*z^5 + 3*z^4 + 4*z^3 + 10*z^2 + 6q^{9} + 2*z^19 + z^18 + 2*z^17 + z^16 + 9*z^14 + 3*z^13 + 4*z^12 + 2*z^11 + 5*z^10 + 10*z^9 + 8*z^8 + 6*z^7 + z^6 + 9*z^5 + 5*z^4 + z^3 + 3*z^2 + 8*z + 3q^{10} + 6*z^19 + 2*z^18 + 8*z^17 + 5*z^16 + 5*z^15 + z^13 + 2*z^12 + 4*z^11 + 6*z^10 + 7*z^9 + 9*z^8 + 3*z^7 + 4*z^6 + z^5 + 9*z^4 + 9*z^3 + z^2 + 6*z + 3q^{12} + 10q^{13} + 3*z^19 + 9*z^18 + 4*z^17 + 7*z^16 + 3*z^15 + 4*z^13 + 7*z^12 + 9*z^11 + 8*z^10 + 6*z^9 + z^8 + 2*z^7 + 7*z^6 + 2*z^5 + 5*z^3 + 2*z^2 + 7q^{14} + z^19 + 3*z^18 + 3*z^17 + 8*z^16 + 8*z^15 + z^14 + 7*z^13 + 9*z^12 + 10*z^10 + 3*z^9 + 6*z^8 + 3*z^7 + z^6 + 4*z^5 + 2*z^2 + 4*z + 9q^{15} + 7*z^19 + 7*z^17 + 6*z^16 + 7*z^15 + 4*z^14 + 10*z^13 + 10*z^12 + 6*z^11 + 5*z^10 + 4*z^9 + z^8 + 6*z^7 + 2*z^6 + 3*z^5 + 4*z^4 + 5*z^3 + 4*z^2 + 8*z + 7q^{16} + 3*z^19 + z^18 + 5*z^16 + 5*z^15 + 8*z^12 + 7*z^11 + 5*z^10 + 6*z^9 + 9*z^8 + 8*z^7 + 2*z^6 + 2*z^5 + 5*z^4 + 6*z^3 + 6*z^2 + 10*zq^{17} + 3*z^19 + 8*z^18 + 6*z^17 + 3*z^16 + 6*z^15 + 9*z^14 + z^12 + 7*z^11 + 7*z^10 + 7*z^9 + 10*z^8 + 9*z^7 + 4*z^6 + 3*z^5 + 7*z^4 + z^3 + 9*z^2 + 7q^{18} + 5*z^19 + 9*z^18 + 8*z^17 + 2*z^16 + 9*z^15 + 8*z^14 + 8*z^13 + 10*z^12 + 9*z^11 + 3*z^9 + 4*z^8 + 4*z^7 + 10*z^5 + 9*z^4 + 6*z^3 + 6*z^2 + 3*z + 7q^{19} + 6*z^19 + 5*z^18 + 3*z^17 + 3*z^16 + z^15 + z^14 + 6*z^13 + 7*z^12 + 6*z^11 + 8*z^10 + 6*z^9 + z^7 + 5*z^6 + 9*z^5 + 7*z^4 + 2*z^3 + 10*z^2 + 10*z + 9q^{20} + 2*z^18 + 5*z^17 + 4*z^16 + 10*z^15 + 8*z^14 + 3*z^13 + 8*z^12 + z^11 + 10*z^10 + z^9 + 8*z^8 + 3*z^7 + 8*z^6 + 10*z^5 + 10*z^4 + 6*z^2 + 2q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^126793q^{2} + z^61740q^{3} + z^49474q^{4} + z^74734q^{5} + z^27483q^{6} + z^7617q^{7} + z^144787q^{8} + z^97237q^{9} + z^40477q^{10} + z^132302q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^28748)\omega^{0}&0\\0&u(z^132302)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(126620356400921773329 + O(11^20))*a^4 + (116124275029971191609 + O(11^20))*a^3 + (54200904031282999190 + O(11^20))*a^2 + (27192497000010582578 + O(11^20))*a - 48718083868206747624 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(141869525449007678538 + O(11^20))*a^4 + (151605097979372508 + O(11^20))*a^3 - (194258345799019839578 + O(11^20))*a^2 - (13567311038568361865 + O(11^20))*a - 192798036755544856348 + O(11^20)q^{5} + -(166098107277332652573 + O(11^20))*a^4 + (262077544250279532283 + O(11^20))*a^3 - (261813859077022280337 + O(11^20))*a^2 + (39649627902202339891 + O(11^20))*a + 28297258544846119286 + O(11^20)q^{6} + -(27523841579649731365 + O(11^20))*a^4 + (20989109331104192790*11 + O(11^20))*a^3 - (116279107353648712844 + O(11^20))*a^2 - (232160258876642163398 + O(11^20))*a + 8449694136397374108 + O(11^20)q^{7} + a^3 - 4194304*aq^{8} + -(206728757606538797473 + O(11^20))*a^4 + (151039960356708325344 + O(11^20))*a^3 - (143140254869584811747 + O(11^20))*a^2 + (158708028880781775585 + O(11^20))*a + 258252730417370027917 + O(11^20)q^{9} + (2347130664133950252 + O(11^20))*a^4 - (273666526199816968253 + O(11^20))*a^3 - (146133167218784058890 + O(11^20))*a^2 - (168630994190763393409 + O(11^20))*a - 77518454593766032615 + O(11^20)q^{10} + (16934259358485004225*11 + O(11^20))*a^4 + (21643924749675957590*11 + O(11^20))*a^3 - (18870650781680015859*11 + O(11^20))*a^2 + (843567987186155850*11^2 + O(11^20))*a - 11614714628517476181*11 + O(11^20)q^{11} + -(39441557776730614930 + O(11^20))*a^4 - (179456845973125848329 + O(11^20))*a^3 - (66925150743570368834 + O(11^20))*a^2 - (245765224208019750989 + O(11^20))*a + 186082390215892588565 + O(11^20)q^{12} + -137858491849q^{13} + -(8392291303721118361 + O(11^20))*a^4 + (94080764106716336602 + O(11^20))*a^3 + (52533650851688319072 + O(11^20))*a^2 + (118488488431490924845 + O(11^20))*a + 17266215589407957762 + O(11^20)q^{14} + (130120802818579978503 + O(11^20))*a^4 + (284867432704604956532 + O(11^20))*a^3 + (132137272138135842803 + O(11^20))*a^2 + (287599407903525171023 + O(11^20))*a - 66108796264582541456 + O(11^20)q^{15} + a^4 - 6291456*a^2 + 4398046511104q^{16} + (247263884779259668634 + O(11^20))*a^4 - (199320934044999770198 + O(11^20))*a^3 + (335640184363866694800 + O(11^20))*a^2 + (111413043909931950062 + O(11^20))*a - 125895449116754295527 + O(11^20)q^{17} + -(129685198089365853755 + O(11^20))*a^4 + (61366251647099302067 + O(11^20))*a^3 - (145396434073872810176 + O(11^20))*a^2 + (289716777941210368284 + O(11^20))*a - 51076396781348042337 + O(11^20)q^{18} + -(102223264689557361929 + O(11^20))*a^4 + (332889877538916840034 + O(11^20))*a^3 + (155178524409514007361 + O(11^20))*a^2 + (236385904455101109948 + O(11^20))*a + 267406116352202105485 + O(11^20)q^{19} + -(106802364945428389007 + O(11^20))*a^4 - (73484333971793824548 + O(11^20))*a^3 + (248452910258062260177 + O(11^20))*a^2 - (237359369857924297976 + O(11^20))*a + 83016675446571761948 + O(11^20)q^{20} + (226558599902903013364 + O(11^20))*a^4 + (306754828431946001816 + O(11^20))*a^3 + (71714185102327232890 + O(11^20))*a^2 + (169717460484611971154 + O(11^20))*a + 26374001414417019471 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + 235939635302487581501 + O(11^20)x^{4} + 19832230278235438204 + O(11^20)x^{3} + -75305933702655388704 + O(11^20)x^{2} + -286001669361950811530 + O(11^20)x + -253175070989686225911 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4*z^19 + 5*z^17 + 5*z^16 + 10*z^15 + 6*z^14 + 3*z^13 + 8*z^12 + z^11 + z^10 + 8*z^9 + 5*z^8 + 2*z^7 + 3*z^6 + 3*z^4 + 3*z^3 + 2*z^2 + 9*z + 9q^{2} + 10*z^19 + 10*z^18 + 9*z^16 + 9*z^15 + 8*z^14 + 9*z^13 + 2*z^12 + 9*z^11 + 10*z^9 + 4*z^8 + 6*z^7 + 2*z^6 + 2*z^5 + 10*z^4 + 2*z^3 + 9*z^2 + 1q^{3} + 3*z^19 + 10*z^18 + 9*z^17 + 10*z^16 + 5*z^15 + z^14 + 8*z^13 + 4*z^12 + 5*z^11 + z^10 + 5*z^9 + 5*z^8 + 7*z^7 + 4*z^6 + 10*z^5 + 2*z^4 + z^3 + z^2 + z + 4q^{4} + 6*z^19 + 8*z^18 + 4*z^17 + z^16 + 7*z^14 + z^13 + 10*z^12 + 6*z^11 + 2*z^10 + 5*z^8 + 3*z^7 + 3*z^6 + 2*z^5 + 3*z^4 + 10*z^3 + 6*z + 10q^{5} + 8*z^19 + 3*z^18 + 10*z^17 + 5*z^16 + z^15 + 4*z^14 + z^13 + 3*z^12 + 6*z^11 + 5*z^10 + 10*z^9 + 10*z^8 + 9*z^7 + 8*z^6 + 9*z^5 + 7*z^4 + z^3 + z^2 + 6*z + 1q^{6} + 5*z^19 + 3*z^17 + 3*z^16 + 2*z^15 + 4*z^13 + 5*z^12 + 6*z^11 + 9*z^10 + 6*z^9 + 8*z^8 + 4*z^7 + 9*z^5 + 5*z^4 + 9*z^3 + 9*z^2 + 7*z + 2q^{7} + z^19 + 2*z^18 + 8*z^17 + 9*z^16 + 5*z^15 + 10*z^14 + 2*z^13 + 6*z^12 + 8*z^11 + 2*z^10 + 9*z^9 + 2*z^8 + z^7 + 9*z^6 + 8*z^5 + 8*z^4 + 10*z^3 + 9*z^2 + 7*z + 9q^{8} + 3*z^19 + 7*z^18 + 4*z^17 + 3*z^16 + z^15 + 9*z^14 + z^12 + 10*z^11 + 6*z^10 + 10*z^9 + z^8 + 6*z^7 + 2*z^6 + 7*z^5 + 3*z^4 + z^3 + 4*z^2 + 8*z + 3q^{9} + 10*z^19 + 5*z^18 + 7*z^17 + 7*z^16 + z^15 + 3*z^14 + 4*z^13 + 5*z^12 + 8*z^10 + 6*z^9 + 3*z^8 + 6*z^7 + 6*z^6 + 5*z^5 + 7*z^4 + 10*z^3 + 3*z^2 + 5*z + 1q^{10} + 8*z^19 + 7*z^18 + 2*z^17 + 5*z^16 + 2*z^15 + 9*z^14 + 2*z^13 + z^12 + 3*z^11 + z^10 + 4*z^9 + 5*z^8 + 3*z^7 + 2*z^6 + 10*z^4 + 5*z^3 + 2*z^2 + 10*z + 8q^{12} + 10q^{13} + z^19 + 9*z^18 + 8*z^17 + 8*z^15 + z^14 + 2*z^13 + 2*z^12 + 10*z^11 + 3*z^10 + 10*z^9 + 6*z^8 + 9*z^7 + 2*z^6 + 6*z^5 + 7*z^4 + 4*z^3 + 10*z^2 + 4*z + 3q^{14} + 5*z^19 + 3*z^18 + 5*z^17 + z^16 + 7*z^15 + 5*z^14 + z^13 + 4*z^12 + 2*z^11 + 9*z^10 + z^9 + 8*z^8 + 3*z^7 + 3*z^6 + 5*z^5 + 8*z^4 + 3*z^3 + 5*z^2 + 9*z + 4q^{15} + 4*z^19 + z^18 + 7*z^17 + 6*z^16 + 8*z^15 + 6*z^13 + 9*z^11 + 8*z^10 + z^9 + 10*z^8 + 5*z^7 + 2*z^6 + 8*z^5 + 3*z^4 + z^3 + 10*z^2 + 9*z + 8q^{16} + 7*z^19 + 5*z^18 + 9*z^17 + 2*z^16 + z^15 + 3*z^14 + 4*z^13 + 8*z^12 + z^11 + 4*z^10 + 8*z^9 + 3*z^8 + z^7 + 6*z^6 + 10*z^5 + 6*z^4 + 4*z^3 + 2*z^2 + 9*z + 10q^{17} + 5*z^19 + 7*z^18 + 6*z^17 + 7*z^16 + 9*z^15 + 9*z^14 + 5*z^13 + 6*z^12 + 7*z^11 + 7*z^10 + 2*z^9 + 3*z^7 + 2*z^6 + 7*z^5 + 6*z^4 + z^3 + z^2 + 5*zq^{18} + 9*z^19 + 3*z^18 + 4*z^17 + 7*z^16 + 7*z^15 + 6*z^13 + 6*z^12 + 5*z^11 + 2*z^10 + 10*z^9 + 6*z^8 + z^7 + 6*z^6 + 10*z^5 + 9*z^4 + 7*z^2 + z + 8q^{19} + 6*z^19 + 4*z^18 + 7*z^17 + 4*z^16 + 3*z^15 + 10*z^13 + 8*z^12 + 7*z^11 + 5*z^10 + z^8 + 2*z^7 + 8*z^6 + 10*z^5 + 8*z^4 + 4*z^3 + 7*zq^{20} + 8*z^19 + 10*z^18 + 5*z^17 + 10*z^16 + 8*z^15 + 4*z^14 + 6*z^12 + 6*z^11 + 5*z^10 + 5*z^9 + 5*z^8 + 10*z^7 + 6*z^5 + 3*z^4 + 4*z^3 + 3*z^2 + 8*z + 5q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^114013q^{2} + z^122740q^{3} + z^106984q^{4} + z^6794q^{5} + z^75703q^{6} + z^73897q^{7} + z^86367q^{8} + z^125967q^{9} + z^120807q^{10} + z^85232q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^75818)\omega^{0}&0\\0&u(z^85232)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(126620356400921773329 + O(11^20))*a^4 + (116124275029971191609 + O(11^20))*a^3 + (54200904031282999190 + O(11^20))*a^2 + (27192497000010582578 + O(11^20))*a - 48718083868206747624 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(141869525449007678538 + O(11^20))*a^4 + (151605097979372508 + O(11^20))*a^3 - (194258345799019839578 + O(11^20))*a^2 - (13567311038568361865 + O(11^20))*a - 192798036755544856348 + O(11^20)q^{5} + -(166098107277332652573 + O(11^20))*a^4 + (262077544250279532283 + O(11^20))*a^3 - (261813859077022280337 + O(11^20))*a^2 + (39649627902202339891 + O(11^20))*a + 28297258544846119286 + O(11^20)q^{6} + -(27523841579649731365 + O(11^20))*a^4 + (20989109331104192790*11 + O(11^20))*a^3 - (116279107353648712844 + O(11^20))*a^2 - (232160258876642163398 + O(11^20))*a + 8449694136397374108 + O(11^20)q^{7} + a^3 - 4194304*aq^{8} + -(206728757606538797473 + O(11^20))*a^4 + (151039960356708325344 + O(11^20))*a^3 - (143140254869584811747 + O(11^20))*a^2 + (158708028880781775585 + O(11^20))*a + 258252730417370027917 + O(11^20)q^{9} + (2347130664133950252 + O(11^20))*a^4 - (273666526199816968253 + O(11^20))*a^3 - (146133167218784058890 + O(11^20))*a^2 - (168630994190763393409 + O(11^20))*a - 77518454593766032615 + O(11^20)q^{10} + (16934259358485004225*11 + O(11^20))*a^4 + (21643924749675957590*11 + O(11^20))*a^3 - (18870650781680015859*11 + O(11^20))*a^2 + (843567987186155850*11^2 + O(11^20))*a - 11614714628517476181*11 + O(11^20)q^{11} + -(39441557776730614930 + O(11^20))*a^4 - (179456845973125848329 + O(11^20))*a^3 - (66925150743570368834 + O(11^20))*a^2 - (245765224208019750989 + O(11^20))*a + 186082390215892588565 + O(11^20)q^{12} + -137858491849q^{13} + -(8392291303721118361 + O(11^20))*a^4 + (94080764106716336602 + O(11^20))*a^3 + (52533650851688319072 + O(11^20))*a^2 + (118488488431490924845 + O(11^20))*a + 17266215589407957762 + O(11^20)q^{14} + (130120802818579978503 + O(11^20))*a^4 + (284867432704604956532 + O(11^20))*a^3 + (132137272138135842803 + O(11^20))*a^2 + (287599407903525171023 + O(11^20))*a - 66108796264582541456 + O(11^20)q^{15} + a^4 - 6291456*a^2 + 4398046511104q^{16} + (247263884779259668634 + O(11^20))*a^4 - (199320934044999770198 + O(11^20))*a^3 + (335640184363866694800 + O(11^20))*a^2 + (111413043909931950062 + O(11^20))*a - 125895449116754295527 + O(11^20)q^{17} + -(129685198089365853755 + O(11^20))*a^4 + (61366251647099302067 + O(11^20))*a^3 - (145396434073872810176 + O(11^20))*a^2 + (289716777941210368284 + O(11^20))*a - 51076396781348042337 + O(11^20)q^{18} + -(102223264689557361929 + O(11^20))*a^4 + (332889877538916840034 + O(11^20))*a^3 + (155178524409514007361 + O(11^20))*a^2 + (236385904455101109948 + O(11^20))*a + 267406116352202105485 + O(11^20)q^{19} + -(106802364945428389007 + O(11^20))*a^4 - (73484333971793824548 + O(11^20))*a^3 + (248452910258062260177 + O(11^20))*a^2 - (237359369857924297976 + O(11^20))*a + 83016675446571761948 + O(11^20)q^{20} + (226558599902903013364 + O(11^20))*a^4 + (306754828431946001816 + O(11^20))*a^3 + (71714185102327232890 + O(11^20))*a^2 + (169717460484611971154 + O(11^20))*a + 26374001414417019471 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{5} + 235939635302487581501 + O(11^20)x^{4} + 19832230278235438204 + O(11^20)x^{3} + -75305933702655388704 + O(11^20)x^{2} + -286001669361950811530 + O(11^20)x + -253175070989686225911 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 6q^{3} + 3q^{4} + 3q^{5} + 9q^{6} + 7q^{8} + 10q^{10} + 7q^{12} + q^{13} + 7q^{15} + 10q^{16} + 4q^{17} + 4q^{19} + 9q^{20} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 9q^{2} + 2q^{3} + 9q^{4} + 5q^{5} + 7q^{6} + 5q^{8} + q^{10} + 3q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{0}&0\\0&u(3)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 262181893417714305427 + O(11^20)q^{2} + -174822783418317580342 + O(11^20)q^{3} + 6771797187940663024 + O(11^20)q^{4} + 155278044844411215334 + O(11^20)q^{5} + 244135640518448845406 + O(11^20)q^{6} + -21212439213639365510*11 + O(11^20)q^{7} + -278283433385866024208 + O(11^20)q^{8} + -14411991429224620334*11 + O(11^20)q^{9} + -184213244744869145565 + O(11^20)q^{10} + 30072391562676185467*11 + O(11^20)q^{11} + -251630010589184376791 + O(11^20)q^{12} + 137858491849q^{13} + 8402159660865650986*11 + O(11^20)q^{14} + -230434857609411452573 + O(11^20)q^{15} + 265255014858109617505 + O(11^20)q^{16} + 155374830379721516425 + O(11^20)q^{17} + -29212630374336327562*11 + O(11^20)q^{18} + -146929748077572279574 + O(11^20)q^{19} + -317229535125808514497 + O(11^20)q^{20} + -13446932166260753333*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6*z^11 + 5*z^10 + 5*z^9 + 7*z^8 + 9*z^7 + 9*z^6 + 10*z^5 + 8*z^4 + 5*z^3 + 3*z^2 + 7*z + 1q^{2} + 3*z^11 + 3*z^10 + 3*z^9 + z^8 + 5*z^7 + 2*z^6 + 2*z^5 + 3*z^4 + 9*z^3 + 8*z^2 + 6*z + 10q^{3} + 9*z^11 + z^10 + z^9 + 10*z^8 + 7*z^7 + 2*z^6 + 10*z^5 + 3*z^4 + 8*z^2 + 8q^{4} + 10*z^11 + 4*z^10 + 4*z^9 + 4*z^8 + 7*z^7 + 6*z^5 + 7*z^3 + z + 6q^{5} + 5*z^11 + 4*z^10 + 4*z^9 + 3*z^8 + z^6 + 2*z^5 + 7*z^4 + 2*z^3 + 4*z^2 + 5*z + 3q^{6} + z^11 + z^10 + z^9 + 4*z^8 + 9*z^7 + 8*z^6 + 8*z^5 + z^4 + 3*z^3 + 10*z^2 + 2*z + 2q^{7} + 5*z^11 + 2*z^10 + 2*z^9 + 2*z^8 + 9*z^7 + 3*z^5 + 9*z^3 + 6*z + 4q^{8} + 4*z^11 + z^10 + z^9 + 9*z^8 + 3*z^6 + 6*z^5 + 10*z^4 + 6*z^3 + z^2 + 4*zq^{9} + 10*z^11 + 5*z^10 + 5*z^9 + 10*z^8 + 8*z^7 + 6*z^6 + 9*z^4 + 9*z^3 + 2*z^2 + 6*z + 7q^{10} + 4*z^11 + 2*z^10 + 2*z^9 + 4*z^8 + z^7 + 9*z^6 + 8*z^4 + 8*z^3 + 3*z^2 + 9*z + 9q^{12} + q^{13} + z^11 + 2*z^10 + 2*z^9 + 10*z^8 + 10*z^7 + 3*z^6 + 2*z^5 + 10*z^4 + 5*z^3 + z^2 + 7*z + 7q^{14} + 6*z^11 + 4*z^10 + 4*z^9 + z^8 + 8*z^7 + 3*z^6 + 5*z^5 + 10*z^4 + 3*z^3 + z^2 + 2*z + 8q^{15} + 4*z^11 + 3*z^8 + 10*z^7 + 8*z^6 + z^5 + z^4 + 4*z^3 + 10*z^2 + 10*z + 5q^{16} + z^11 + 5*z^10 + 5*z^9 + 6*z^8 + 2*z^7 + 10*z^6 + 6*z^5 + 4*z^4 + 7*z^2 + 1q^{17} + 9*z^11 + z^10 + z^9 + 10*z^8 + 7*z^7 + 2*z^6 + 10*z^5 + 3*z^4 + 8*z^2 + 1q^{18} + 4*z^11 + 7*z^10 + 7*z^9 + z^8 + 6*z^7 + 6*z^6 + 3*z^5 + 9*z^4 + 7*z^3 + 2*z^2 + z + 3q^{19} + 4*z^11 + 2*z^10 + 2*z^9 + 4*z^8 + z^7 + 9*z^6 + 8*z^4 + 8*z^3 + 3*z^2 + 9*z + 6q^{20} + z^11 + 9*z^8 + 8*z^7 + 2*z^6 + 3*z^5 + 3*z^4 + z^3 + 8*z^2 + 8*z + 6q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^869q^{2} + z^825q^{3} + z^1223q^{4} + z^819q^{5} + z^364q^{6} + z^1257q^{7} + z^1122q^{8} + z^277q^{9} + z^358q^{10} + z^684q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^646)\omega^{0}&0\\0&u(z^684)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(228494176128019091054 + O(11^20))*a^2 + (126232126130457204851 + O(11^20))*a + 1007663414535709320*11 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(95366962654606477607 + O(11^20))*a^2 + (286975885151558282622 + O(11^20))*a - 311941616959375706649 + O(11^20)q^{5} + -(283882396171998047147 + O(11^20))*a^2 - (213752326028574652862 + O(11^20))*a + 196019062370748611157 + O(11^20)q^{6} + (149635348179479808402 + O(11^20))*a^2 - (329976420198962930333 + O(11^20))*a - 223322087805285745160 + O(11^20)q^{7} + -(18829220815627542624 + O(11^20))*a^2 - (207662230975403199057 + O(11^20))*a - 68567498352877310860 + O(11^20)q^{8} + -(72550889623189489209 + O(11^20))*a^2 - (9542890798164020166 + O(11^20))*a - 263903860454195927970 + O(11^20)q^{9} + -(257941820253429481272 + O(11^20))*a^2 + (71089598886485630521 + O(11^20))*a + 268333037649251759590 + O(11^20)q^{10} + (30167737553010492191*11 + O(11^20))*a^2 + (17609105808763440729*11 + O(11^20))*a - 8809998171776527779*11 + O(11^20)q^{11} + -(302101307293963381738 + O(11^20))*a^2 - (138292184510539719423 + O(11^20))*a - 287404695168779539376 + O(11^20)q^{12} + 137858491849q^{13} + -(15435291878799267170 + O(11^20))*a^2 - (29332476439212035770 + O(11^20))*a + 14271130288616791857 + O(11^20)q^{14} + -(209594106586688373846 + O(11^20))*a^2 - (216960595009583517615 + O(11^20))*a - 333513083638291995489 + O(11^20)q^{15} + (79696005886396998655 + O(11^20))*a^2 + (160632323172559380816 + O(11^20))*a + 138839897361701815992 + O(11^20)q^{16} + -(190492124401294766397 + O(11^20))*a^2 - (5148513524370172444*11 + O(11^20))*a + 18894816919005700998 + O(11^20)q^{17} + (264441398572581411358 + O(11^20))*a^2 - (21067508477419604608*11 + O(11^20))*a + 296698086672838798004 + O(11^20)q^{18} + -(23941721301908660132*11 + O(11^20))*a^2 - (72314759768821085937 + O(11^20))*a + 193621124714059036077 + O(11^20)q^{19} + -(203072463007584879990 + O(11^20))*a^2 - (138593765599293151966 + O(11^20))*a - 221831975660220712232 + O(11^20)q^{20} + -(161604655615340483558 + O(11^20))*a^2 - (287500465515811730987 + O(11^20))*a + 156087460811032487425 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 18829220815627542624 + O(11^20)x^{2} + 207662230975399004753 + O(11^20)x + 68567498352877310860 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5*z^11 + 7*z^10 + 7*z^9 + 10*z^8 + 3*z^7 + 8*z^6 + 6*z^5 + z^4 + 8*z^3 + 10*z^2 + 9*zq^{2} + 5*z^11 + z^10 + z^9 + 7*z^8 + 8*z^7 + 5*z^6 + 9*z^5 + 2*z^4 + 7*z^3 + 9*z^2 + z + 5q^{3} + 8*z^11 + 10*z^10 + 10*z^9 + 8*z^7 + 10*z^6 + 8*z^5 + 4*z^4 + 6*z^3 + 7*z^2 + 4*z + 10q^{4} + 5*z^11 + 4*z^10 + 4*z^9 + 3*z^8 + z^6 + 2*z^5 + 7*z^4 + 2*z^3 + 4*z^2 + 5*z + 3q^{5} + 7*z^11 + 3*z^10 + 3*z^9 + 4*z^8 + 4*z^7 + 10*z^6 + 3*z^5 + 4*z^4 + 2*z^3 + 7*z^2 + 5*zq^{6} + 9*z^11 + 4*z^10 + 4*z^9 + 6*z^8 + 10*z^7 + 9*z^6 + 3*z^5 + 8*z^4 + 6*z^3 + 3*z^2 + 4*z + 4q^{7} + 8*z^11 + 2*z^10 + 2*z^9 + 7*z^8 + 6*z^6 + z^5 + 9*z^4 + z^3 + 2*z^2 + 8*z + 8q^{8} + 10*z^11 + 9*z^10 + 9*z^9 + z^8 + z^7 + 8*z^6 + 9*z^5 + z^4 + 6*z^3 + 10*z^2 + 4*z + 2q^{9} + 10*z^11 + 10*z^10 + 10*z^9 + 7*z^8 + 2*z^7 + 3*z^6 + 3*z^5 + 10*z^4 + 8*z^3 + z^2 + 9*z + 6q^{10} + 4*z^11 + 4*z^10 + 4*z^9 + 5*z^8 + 3*z^7 + 10*z^6 + 10*z^5 + 4*z^4 + z^3 + 7*z^2 + 8*z + 2q^{12} + q^{13} + 3*z^11 + 10*z^10 + 10*z^9 + 10*z^8 + z^7 + 4*z^5 + z^3 + 8*zq^{14} + z^10 + z^9 + 6*z^8 + z^7 + 6*z^6 + 5*z^5 + 9*z^4 + 2*z^3 + 2*z^2 + 5*z + 5q^{15} + 5*z^11 + 3*z^10 + 3*z^9 + 8*z^8 + 10*z^7 + 6*z^6 + 8*z^5 + 9*z^4 + 2*z^2 + 5q^{16} + 7*z^11 + 6*z^10 + 6*z^9 + 7*z^7 + 6*z^6 + 7*z^5 + 9*z^4 + 8*z^3 + 2*z^2 + 9*zq^{17} + 8*z^11 + 10*z^10 + 10*z^9 + 8*z^7 + 10*z^6 + 8*z^5 + 4*z^4 + 6*z^3 + 7*z^2 + 4*z + 3q^{18} + 7*z^11 + z^10 + z^9 + 3*z^8 + 2*z^7 + 9*z^6 + 4*z^5 + 8*z^4 + 9*z^3 + 3*z^2 + 6*z + 6q^{19} + 4*z^11 + 4*z^10 + 4*z^9 + 5*z^8 + 3*z^7 + 10*z^6 + 10*z^5 + 4*z^4 + z^3 + 7*z^2 + 8*z + 10q^{20} + 4*z^11 + 9*z^10 + 9*z^9 + 2*z^8 + 8*z^7 + 7*z^6 + 2*z^5 + 5*z^4 + 6*z^2 + 6q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^249q^{2} + z^1095q^{3} + z^153q^{4} + z^1029q^{5} + z^14q^{6} + z^527q^{7} + z^372q^{8} + z^387q^{9} + z^1278q^{10} + z^874q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^456)\omega^{0}&0\\0&u(z^874)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(228494176128019091054 + O(11^20))*a^2 + (126232126130457204851 + O(11^20))*a + 1007663414535709320*11 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(95366962654606477607 + O(11^20))*a^2 + (286975885151558282622 + O(11^20))*a - 311941616959375706649 + O(11^20)q^{5} + -(283882396171998047147 + O(11^20))*a^2 - (213752326028574652862 + O(11^20))*a + 196019062370748611157 + O(11^20)q^{6} + (149635348179479808402 + O(11^20))*a^2 - (329976420198962930333 + O(11^20))*a - 223322087805285745160 + O(11^20)q^{7} + -(18829220815627542624 + O(11^20))*a^2 - (207662230975403199057 + O(11^20))*a - 68567498352877310860 + O(11^20)q^{8} + -(72550889623189489209 + O(11^20))*a^2 - (9542890798164020166 + O(11^20))*a - 263903860454195927970 + O(11^20)q^{9} + -(257941820253429481272 + O(11^20))*a^2 + (71089598886485630521 + O(11^20))*a + 268333037649251759590 + O(11^20)q^{10} + (30167737553010492191*11 + O(11^20))*a^2 + (17609105808763440729*11 + O(11^20))*a - 8809998171776527779*11 + O(11^20)q^{11} + -(302101307293963381738 + O(11^20))*a^2 - (138292184510539719423 + O(11^20))*a - 287404695168779539376 + O(11^20)q^{12} + 137858491849q^{13} + -(15435291878799267170 + O(11^20))*a^2 - (29332476439212035770 + O(11^20))*a + 14271130288616791857 + O(11^20)q^{14} + -(209594106586688373846 + O(11^20))*a^2 - (216960595009583517615 + O(11^20))*a - 333513083638291995489 + O(11^20)q^{15} + (79696005886396998655 + O(11^20))*a^2 + (160632323172559380816 + O(11^20))*a + 138839897361701815992 + O(11^20)q^{16} + -(190492124401294766397 + O(11^20))*a^2 - (5148513524370172444*11 + O(11^20))*a + 18894816919005700998 + O(11^20)q^{17} + (264441398572581411358 + O(11^20))*a^2 - (21067508477419604608*11 + O(11^20))*a + 296698086672838798004 + O(11^20)q^{18} + -(23941721301908660132*11 + O(11^20))*a^2 - (72314759768821085937 + O(11^20))*a + 193621124714059036077 + O(11^20)q^{19} + -(203072463007584879990 + O(11^20))*a^2 - (138593765599293151966 + O(11^20))*a - 221831975660220712232 + O(11^20)q^{20} + -(161604655615340483558 + O(11^20))*a^2 - (287500465515811730987 + O(11^20))*a + 156087460811032487425 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 18829220815627542624 + O(11^20)x^{2} + 207662230975399004753 + O(11^20)x + 68567498352877310860 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10*z^10 + 10*z^9 + 5*z^8 + 10*z^7 + 5*z^6 + 6*z^5 + 2*z^4 + 9*z^3 + 9*z^2 + 6*z + 3q^{2} + 3*z^11 + 7*z^10 + 7*z^9 + 3*z^8 + 9*z^7 + 4*z^6 + 6*z^4 + 6*z^3 + 5*z^2 + 4*z + 7q^{3} + 5*z^11 + z^8 + 7*z^7 + 10*z^6 + 4*z^5 + 4*z^4 + 5*z^3 + 7*z^2 + 7*z + 8q^{4} + 7*z^11 + 3*z^10 + 3*z^9 + 4*z^8 + 4*z^7 + 10*z^6 + 3*z^5 + 4*z^4 + 2*z^3 + 7*z^2 + 5*zq^{5} + 10*z^11 + 4*z^10 + 4*z^9 + 4*z^8 + 7*z^7 + 6*z^5 + 7*z^3 + z + 6q^{6} + z^11 + 6*z^10 + 6*z^9 + z^8 + 3*z^7 + 5*z^6 + 2*z^4 + 2*z^3 + 9*z^2 + 5*z + 1q^{7} + 9*z^11 + 7*z^10 + 7*z^9 + 2*z^8 + 2*z^7 + 5*z^6 + 7*z^5 + 2*z^4 + z^3 + 9*z^2 + 8*z + 1q^{8} + 8*z^11 + z^10 + z^9 + z^8 + 10*z^7 + 7*z^5 + 10*z^3 + 3*z + 9q^{9} + 2*z^11 + 7*z^10 + 7*z^9 + 5*z^8 + z^7 + 2*z^6 + 8*z^5 + 3*z^4 + 5*z^3 + 8*z^2 + 7*z + 4q^{10} + 3*z^11 + 5*z^10 + 5*z^9 + 2*z^8 + 7*z^7 + 3*z^6 + z^5 + 10*z^4 + 2*z^3 + z^2 + 5*z + 10q^{12} + q^{13} + 7*z^11 + 10*z^10 + 10*z^9 + 2*z^8 + 8*z^6 + 5*z^5 + z^4 + 5*z^3 + 10*z^2 + 7*z + 9q^{14} + 5*z^11 + 6*z^10 + 6*z^9 + 4*z^8 + 2*z^7 + 2*z^6 + z^5 + 3*z^4 + 6*z^3 + 8*z^2 + 4*z + 7q^{15} + 2*z^11 + 8*z^10 + 8*z^9 + 2*z^7 + 8*z^6 + 2*z^5 + z^4 + 7*z^3 + 10*z^2 + zq^{16} + 3*z^11 + 5*z^8 + 2*z^7 + 6*z^6 + 9*z^5 + 9*z^4 + 3*z^3 + 2*z^2 + 2*z + 1q^{17} + 5*z^11 + z^8 + 7*z^7 + 10*z^6 + 4*z^5 + 4*z^4 + 5*z^3 + 7*z^2 + 7*z + 1q^{18} + 3*z^10 + 3*z^9 + 7*z^8 + 3*z^7 + 7*z^6 + 4*z^5 + 5*z^4 + 6*z^3 + 6*z^2 + 4*z + 8q^{19} + 3*z^11 + 5*z^10 + 5*z^9 + 2*z^8 + 7*z^7 + 3*z^6 + z^5 + 10*z^4 + 2*z^3 + z^2 + 5*z + 7q^{20} + 6*z^11 + 2*z^10 + 2*z^9 + 6*z^7 + 2*z^6 + 6*z^5 + 3*z^4 + 10*z^3 + 8*z^2 + 3*z + 2q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^79q^{2} + z^75q^{3} + z^353q^{4} + z^679q^{5} + z^154q^{6} + z^477q^{7} + z^102q^{8} + z^267q^{9} + z^758q^{10} + z^304q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1026)\omega^{0}&0\\0&u(z^304)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(228494176128019091054 + O(11^20))*a^2 + (126232126130457204851 + O(11^20))*a + 1007663414535709320*11 + O(11^20)q^{3} + a^2 - 2097152q^{4} + -(95366962654606477607 + O(11^20))*a^2 + (286975885151558282622 + O(11^20))*a - 311941616959375706649 + O(11^20)q^{5} + -(283882396171998047147 + O(11^20))*a^2 - (213752326028574652862 + O(11^20))*a + 196019062370748611157 + O(11^20)q^{6} + (149635348179479808402 + O(11^20))*a^2 - (329976420198962930333 + O(11^20))*a - 223322087805285745160 + O(11^20)q^{7} + -(18829220815627542624 + O(11^20))*a^2 - (207662230975403199057 + O(11^20))*a - 68567498352877310860 + O(11^20)q^{8} + -(72550889623189489209 + O(11^20))*a^2 - (9542890798164020166 + O(11^20))*a - 263903860454195927970 + O(11^20)q^{9} + -(257941820253429481272 + O(11^20))*a^2 + (71089598886485630521 + O(11^20))*a + 268333037649251759590 + O(11^20)q^{10} + (30167737553010492191*11 + O(11^20))*a^2 + (17609105808763440729*11 + O(11^20))*a - 8809998171776527779*11 + O(11^20)q^{11} + -(302101307293963381738 + O(11^20))*a^2 - (138292184510539719423 + O(11^20))*a - 287404695168779539376 + O(11^20)q^{12} + 137858491849q^{13} + -(15435291878799267170 + O(11^20))*a^2 - (29332476439212035770 + O(11^20))*a + 14271130288616791857 + O(11^20)q^{14} + -(209594106586688373846 + O(11^20))*a^2 - (216960595009583517615 + O(11^20))*a - 333513083638291995489 + O(11^20)q^{15} + (79696005886396998655 + O(11^20))*a^2 + (160632323172559380816 + O(11^20))*a + 138839897361701815992 + O(11^20)q^{16} + -(190492124401294766397 + O(11^20))*a^2 - (5148513524370172444*11 + O(11^20))*a + 18894816919005700998 + O(11^20)q^{17} + (264441398572581411358 + O(11^20))*a^2 - (21067508477419604608*11 + O(11^20))*a + 296698086672838798004 + O(11^20)q^{18} + -(23941721301908660132*11 + O(11^20))*a^2 - (72314759768821085937 + O(11^20))*a + 193621124714059036077 + O(11^20)q^{19} + -(203072463007584879990 + O(11^20))*a^2 - (138593765599293151966 + O(11^20))*a - 221831975660220712232 + O(11^20)q^{20} + -(161604655615340483558 + O(11^20))*a^2 - (287500465515811730987 + O(11^20))*a + 156087460811032487425 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 18829220815627542624 + O(11^20)x^{2} + 207662230975399004753 + O(11^20)x + 68567498352877310860 + O(11^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 6q^{2} + 4q^{3} + 6q^{4} + 4q^{5} + 2q^{6} + 9q^{7} + 10q^{8} + 2q^{10} + 7q^{11} + 2q^{12} + 2q^{13} + 10q^{14} + 5q^{15} + q^{16} + 6q^{19} + 2q^{20} + 3q^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + q^{3} + 6q^{4} + 8q^{5} + 6q^{6} + q^{7} + 10q^{8} + 7q^{9} + 4q^{10} + 6q^{12} + 2q^{13} + 6q^{14} + 8q^{15} + q^{16} + 9q^{17} + 9q^{18} + 6q^{19} + 4q^{20} + q^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 5q^{2} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*zq^{3} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 1q^{4} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 3q^{5} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 4q^{6} + q^{7} + 4q^{8} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 5q^{9} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 10q^{10} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 6q^{11} + z^9 + 3*z^7 + 3*z^6 + 6*z^5 + 2*z^2 + 2*z + 4q^{12} + 9q^{13} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 5q^{14} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 4q^{15} + 9*z^9 + 5*z^7 + 5*z^6 + 10*z^5 + 7*z^2 + 7*z + 1q^{16} + 8*z^9 + 2*z^7 + 2*z^6 + 4*z^5 + 5*z^2 + 5*z + 10q^{17} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 1q^{18} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 1q^{19} + 3*z^9 + 9*z^7 + 9*z^6 + 7*z^5 + 6*z^2 + 6*z + 9q^{20} + 4*z^9 + z^7 + z^6 + 2*z^5 + 8*z^2 + 8*zq^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 7q^{2} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 5q^{3} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 3q^{4} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 5q^{5} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 6q^{6} + q^{7} + 4q^{8} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 8q^{9} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 8q^{10} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 8q^{11} + 10*z^9 + 8*z^7 + 8*z^6 + 5*z^5 + 9*z^2 + 9*z + 8q^{12} + 9q^{13} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 7q^{14} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 7q^{15} + 2*z^9 + 6*z^7 + 6*z^6 + z^5 + 4*z^2 + 4*z + 4q^{16} + 3*z^9 + 9*z^7 + 9*z^6 + 7*z^5 + 6*z^2 + 6*z + 9q^{17} + 6*z^9 + 7*z^7 + 7*z^6 + 3*z^5 + z^2 + z + 10q^{18} + 5*z^9 + 4*z^7 + 4*z^6 + 8*z^5 + 10*z^2 + 10*z + 3q^{19} + 8*z^9 + 2*z^7 + 2*z^6 + 4*z^5 + 5*z^2 + 5*z + 10q^{20} + 7*z^9 + 10*z^7 + 10*z^6 + 9*z^5 + 3*z^2 + 3*z + 5q^{21} + \cdots \in S_{14}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^101q^{2} + z^35q^{3} + z^2q^{4} + z^55q^{5} + z^16q^{6} + z^58q^{7} + z^70q^{8} + z^83q^{9} + 8q^{10} + z^115q^{11} + z^37q^{12} + 7q^{13} + z^39q^{14} + z^90q^{15} + z^116q^{16} + z^40q^{17} + z^64q^{18} + z^104q^{19} + z^57q^{20} + z^93q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^31q^{2} + z^25q^{3} + z^22q^{4} + z^5q^{5} + z^56q^{6} + z^38q^{7} + z^50q^{8} + z^73q^{9} + 8q^{10} + z^65q^{11} + z^47q^{12} + 7q^{13} + z^69q^{14} + z^30q^{15} + z^76q^{16} + z^80q^{17} + z^104q^{18} + z^64q^{19} + z^27q^{20} + z^63q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^1067q^{2} + z^752q^{3} + z^696q^{4} + z^839q^{5} + z^489q^{6} + z^361q^{7} + z^318q^{8} + z^56q^{9} + z^576q^{10} + z^517q^{11} + z^118q^{12} + 4q^{13} + z^98q^{14} + z^261q^{15} + z^122q^{16} + z^902q^{17} + z^1123q^{18} + z^794q^{19} + z^205q^{20} + z^1113q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^1097q^{2} + z^292q^{3} + z^1006q^{4} + z^1249q^{5} + z^59q^{6} + z^1311q^{7} + z^838q^{8} + z^616q^{9} + z^1016q^{10} + z^367q^{11} + z^1298q^{12} + 4q^{13} + z^1078q^{14} + z^211q^{15} + z^12q^{16} + z^612q^{17} + z^383q^{18} + z^754q^{19} + z^925q^{20} + z^273q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^97q^{2} + z^552q^{3} + z^426q^{4} + z^439q^{5} + z^649q^{6} + z^1121q^{7} + z^1238q^{8} + z^126q^{9} + z^536q^{10} + z^47q^{11} + z^978q^{12} + 4q^{13} + z^1218q^{14} + z^991q^{15} + z^132q^{16} + z^82q^{17} + z^223q^{18} + z^314q^{19} + z^865q^{20} + z^343q^{21} + \cdots \in S_{16}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + 4q^{3} + 5q^{4} + 2q^{5} + 7q^{6} + 9q^{7} + 2q^{8} + 7q^{9} + 9q^{10} + 10q^{11} + 9q^{12} + 8q^{13} + 2q^{14} + 8q^{15} + 7q^{16} + 4q^{18} + 4q^{19} + 10q^{20} + 3q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 4q^{2} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*z + 4q^{3} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 10q^{4} + 5q^{5} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 2q^{6} + 4*z^5 + 2*z^4 + 4*z^3 + 8*z^2 + z + 7q^{7} + 4q^{8} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*z + 5q^{9} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 9q^{10} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 1q^{11} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 9q^{12} + 8q^{13} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 8q^{14} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 9q^{15} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 1q^{16} + 2*z^5 + z^4 + 2*z^3 + 4*z^2 + 6*z + 5q^{17} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*z + 6q^{18} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 5q^{19} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 6q^{20} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*zq^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*z + 10q^{2} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 8q^{3} + 5*z^5 + 8*z^4 + 5*z^3 + 10*z^2 + 4*z + 6q^{4} + 5q^{5} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*zq^{6} + 7*z^5 + 9*z^4 + 7*z^3 + 3*z^2 + 10*z + 8q^{7} + 4q^{8} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 9q^{9} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 6q^{10} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 4q^{11} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 6q^{12} + 8q^{13} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 5q^{14} + 8*z^5 + 4*z^4 + 8*z^3 + 5*z^2 + 2*z + 7q^{15} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 9q^{16} + 9*z^5 + 10*z^4 + 9*z^3 + 7*z^2 + 5*zq^{17} + 6*z^5 + 3*z^4 + 6*z^3 + z^2 + 7*z + 10q^{18} + z^5 + 6*z^4 + z^3 + 2*z^2 + 3*z + 2q^{19} + 3*z^5 + 7*z^4 + 3*z^3 + 6*z^2 + 9*z + 8q^{20} + 10*z^5 + 5*z^4 + 10*z^3 + 9*z^2 + 8*z + 3q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5*z^5 + 5*z^4 + 5*z^3 + 10*z + 2q^{2} + 9*z^5 + 10*z^4 + 10*z^3 + 3*z^2 + 6*zq^{3} + 3*z^5 + 7*z^4 + 7*z^3 + z^2 + 2*z + 7q^{4} + 10*z^5 + 6*z^4 + 6*z^3 + 10*z^2 + 2*z + 9q^{5} + 2*z^5 + 9*z^4 + 9*z^3 + 10*z^2 + 8*zq^{6} + 5*z^5 + 3*z^4 + 3*z^3 + 5*z^2 + z + 3q^{7} + 8*z^5 + 3*z^4 + 3*z^3 + 7*z^2 + 10*zq^{8} + 6*z^5 + 2*z^4 + 2*z^3 + 10*z^2 + 5*z + 10q^{9} + 10*z^5 + 10*z^4 + 10*z^3 + 9*zq^{10} + z^4 + z^3 + 3*z^2 + 10*z + 1q^{11} + 10*z^5 + z^4 + z^3 + 6*z^2 + 7*z + 10q^{12} + 3q^{13} + 3*z^5 + 3*z^4 + 3*z^3 + 6*z + 8q^{14} + 5*z^4 + 5*z^3 + 4*z^2 + 6*z + 1q^{15} + 7*z^5 + 4*z^4 + 4*z^3 + 2*z^2 + 6*z + 2q^{16} + z^5 + 6*z^4 + 6*z^3 + 4*z^2 + 8*z + 7q^{17} + 3*z^5 + 2*z^4 + 2*z^3 + 8*z^2 + 7*z + 5q^{18} + 9*z^5 + 2*z^4 + 2*z^3 + z^2 + 3*zq^{19} + 4*z^5 + 7*z^4 + 7*z^3 + 9*z^2 + 5*z + 1q^{20} + 3*z^5 + 4*z^4 + 4*z^3 + 3*z^2 + 5*z + 6q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8*z^5 + 3*z^4 + 3*z^3 + 7*z^2 + 10*z + 7q^{2} + 9*z^5 + 4*z^4 + 4*z^3 + 7*z^2 + z + 8q^{3} + 3*z^5 + 5*z^4 + 5*z^3 + 6*z^2 + 4*z + 6q^{4} + 9*z^5 + 5*z^4 + 5*z^3 + 10*z^2 + 1q^{5} + 5*z^5 + 9*z^4 + 9*z^3 + z^2 + 6*z + 6q^{6} + 10*z^5 + 8*z^4 + 8*z^3 + 5*z^2 + 10q^{7} + 9*z^5 + 3*z^4 + 3*z^3 + 4*z^2 + 2*z + 2q^{8} + 7*z^5 + 7*z^4 + 7*z^3 + 3*z + 9q^{9} + 5*z^5 + 6*z^4 + 6*z^3 + 3*z^2 + 9*z + 10q^{10} + 10*z^5 + 3*z^4 + 3*z^3 + z^2 + 5*zq^{11} + 3*z^5 + z^4 + z^3 + 5*z^2 + 8*z + 7q^{12} + 3q^{13} + 7*z^5 + 4*z^4 + 4*z^3 + 2*z^2 + 6*zq^{14} + 6*z^5 + 4*z^4 + 4*z^3 + 5*z^2 + 3*z + 7q^{15} + z^5 + 4*z^4 + 4*z^3 + 9*z^2 + 10*z + 1q^{16} + z^5 + 9*z^4 + 9*z^3 + 2*z^2 + 5*z + 3q^{17} + 8*z^5 + z^4 + z^3 + z^2 + z + 9q^{18} + 6*z^5 + 2*z^4 + 2*z^3 + 10*z^2 + 5*z + 5q^{19} + 10*z^5 + 7*z^4 + 7*z^3 + 2*z^2 + z + 2q^{20} + 6*z^5 + 7*z^4 + 7*z^3 + 3*z^2 + 8q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 9*z^5 + 3*z^4 + 3*z^3 + 4*z^2 + 2*z + 9q^{2} + 4*z^5 + 8*z^4 + 8*z^3 + z^2 + 4*zq^{3} + 5*z^5 + 10*z^4 + 10*z^3 + 4*z^2 + 5*z + 7q^{4} + 3*z^5 + 2*z^2 + 9*z + 3q^{5} + 4*z^5 + 4*z^4 + 4*z^3 + 8*z + 7q^{6} + 7*z^5 + z^2 + 10*zq^{7} + 5*z^5 + 5*z^4 + 5*z^3 + 10*z + 6q^{8} + 9*z^5 + 2*z^4 + 2*z^3 + z^2 + 3*z + 5q^{9} + 7*z^5 + 6*z^4 + 6*z^3 + 8*z^2 + 4*z + 3q^{10} + z^5 + 7*z^4 + 7*z^3 + 7*z^2 + 7*z + 6q^{11} + 9*z^5 + 9*z^4 + 9*z^3 + 7*z + 1q^{12} + 3q^{13} + z^5 + 4*z^4 + 4*z^3 + 9*z^2 + 10*z + 10q^{14} + 5*z^5 + 2*z^4 + 2*z^3 + 2*z^2 + 2*z + 4q^{15} + 3*z^5 + 3*z^4 + 3*z^3 + 6*z + 10q^{16} + 9*z^5 + 7*z^4 + 7*z^3 + 5*z^2 + 9*z + 7q^{17} + 8*z^4 + 8*z^3 + 2*z^2 + 3*z + 2q^{18} + 7*z^5 + 7*z^4 + 7*z^3 + 3*z + 4q^{19} + 8*z^5 + 8*z^4 + 8*z^3 + 5*z + 4q^{20} + 2*z^5 + 5*z^2 + 6*z + 2q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + 6q^{3} + 2q^{4} + 3q^{5} + 4q^{6} + 7q^{7} + 4q^{8} + 5q^{9} + 2q^{10} + 3q^{11} + q^{12} + 3q^{13} + q^{14} + 7q^{15} + 7q^{16} + 5q^{17} + 7q^{18} + 8q^{19} + 6q^{20} + 9q^{21} + \cdots \in S_{18}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 9q^{2} + 2q^{3} + 9q^{4} + 5q^{5} + 7q^{6} + 5q^{8} + q^{10} + 3q^{11} + 7q^{12} + 6q^{13} + 10q^{15} + 2q^{16} + 8q^{17} + 6q^{19} + q^{20} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^869q^{2} + z^825q^{3} + z^1223q^{4} + z^819q^{5} + z^364q^{6} + z^1257q^{7} + z^1122q^{8} + z^277q^{9} + z^358q^{10} + z^684q^{11} + z^718q^{12} + 6q^{13} + z^796q^{14} + z^314q^{15} + z^554q^{16} + z^907q^{17} + z^1146q^{18} + z^1099q^{19} + z^712q^{20} + z^752q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^79q^{2} + z^75q^{3} + z^353q^{4} + z^679q^{5} + z^154q^{6} + z^477q^{7} + z^102q^{8} + z^267q^{9} + z^758q^{10} + z^304q^{11} + z^428q^{12} + 6q^{13} + z^556q^{14} + z^754q^{15} + z^534q^{16} + z^687q^{17} + z^346q^{18} + z^1309q^{19} + z^1032q^{20} + z^552q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^249q^{2} + z^1095q^{3} + z^153q^{4} + z^1029q^{5} + z^14q^{6} + z^527q^{7} + z^372q^{8} + z^387q^{9} + z^1278q^{10} + z^874q^{11} + z^1248q^{12} + 6q^{13} + z^776q^{14} + z^794q^{15} + z^774q^{16} + z^667q^{17} + z^636q^{18} + z^119q^{19} + z^1182q^{20} + z^292q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^9 + 8*z^8 + 10*z^7 + 10*z^6 + 7*z^5 + 8*z^4 + 2*z^3 + 2*z^2 + 8*z + 10q^{2} + 9*z^9 + 9*z^8 + z^7 + z^6 + 5*z^4 + 10*z^3 + 3*z^2 + 10*z + 6q^{3} + 2*z^8 + 4*z^7 + 9*z^6 + 5*z^5 + 9*z^4 + 8*z^3 + 8*z^2 + 9*zq^{4} + 7*z^9 + 3*z^8 + 6*z^7 + 5*z^5 + 2*z^4 + z^3 + 7*z^2 + 9*z + 7q^{5} + 4*z^9 + 7*z^8 + 3*z^7 + 7*z^6 + 3*z^5 + 8*z^4 + 5*z^3 + 6*z^2 + 10*z + 6q^{6} + 6*z^9 + z^8 + 5*z^7 + 8*z^6 + 6*z^5 + 9*z^3 + 6*z^2 + 3*z + 2q^{7} + 10*z^9 + z^8 + 9*z^7 + 5*z^6 + 6*z^5 + 5*z^4 + 4*z^3 + 4*z^2 + 3*z + 1q^{8} + 8*z^9 + z^8 + 7*z^7 + 6*z^6 + z^5 + 10*z^4 + 2*z^3 + 10*z^2 + 7*z + 7q^{9} + 9*z^9 + 8*z^8 + 10*z^7 + 10*z^6 + 6*z^5 + 7*z^4 + z^3 + 7*z^2 + 2*zq^{10} + 8*z^8 + 5*z^7 + 5*z^6 + 7*z^5 + 6*z^4 + 6*z^3 + z^2 + 9*z + 6q^{11} + 8*z^9 + 5*z^8 + 2*z^7 + 9*z^6 + 8*z^5 + 6*z^4 + z^3 + 9*z^2 + 7*z + 3q^{12} + 5q^{13} + 3*z^9 + 10*z^8 + 3*z^6 + 9*z^5 + 10*z^4 + 7*z^3 + 6*z^2 + 8*zq^{14} + 10*z^9 + 2*z^8 + 3*z^7 + 8*z^6 + 4*z^5 + z^4 + 10*z^3 + 4*z^2 + z + 7q^{15} + z^8 + 10*z^7 + 10*z^6 + 3*z^5 + 8*z^4 + 5*z^3 + z^2 + z + 5q^{16} + 10*z^9 + 2*z^8 + 10*z^7 + 10*z^6 + 10*z^5 + 5*z^4 + 2*z^2 + 6*z + 8q^{17} + 7*z^9 + 2*z^8 + z^7 + 7*z^6 + 8*z^5 + 10*z^4 + 8*z^3 + 6*z^2 + 8*z + 8q^{18} + 2*z^9 + 3*z^8 + 5*z^7 + 4*z^6 + 5*z^5 + 2*z^4 + 10*z^3 + 8*z + 4q^{19} + 3*z^9 + 8*z^8 + 2*z^7 + 3*z^6 + 5*z^5 + 2*z^4 + z^3 + 2*z^2q^{20} + 5*z^9 + 4*z^8 + 5*z^7 + 6*z^6 + 4*z^5 + 7*z^4 + 6*z^3 + 5*z + 5q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8*z^8 + 4*z^7 + 4*z^6 + 10*z^5 + 2*z^4 + z^3 + 10*z^2 + 3*z + 7q^{2} + 7*z^8 + 9*z^7 + 8*z^6 + 7*z^5 + 3*z^4 + 7*z^3 + 5*z^2 + z + 8q^{3} + 4*z^9 + 4*z^8 + 2*z^7 + 3*z^6 + 9*z^5 + 2*z^4 + 9*z^3 + 3*z^2 + 7*z + 9q^{4} + 6*z^9 + 5*z^8 + z^6 + z^5 + 3*z^4 + 6*z^3 + 6*z^2 + 6*z + 4q^{5} + 9*z^9 + 8*z^8 + 5*z^7 + 4*z^5 + 7*z^4 + 8*z^3 + 3*z^2 + z + 3q^{6} + 4*z^9 + 5*z^8 + 6*z^7 + 8*z^6 + 7*z^5 + 4*z^4 + 4*z^3 + 4*z^2 + 8*zq^{7} + 7*z^9 + 7*z^8 + 10*z^7 + 2*z^6 + 6*z^5 + 8*z^4 + 10*z^3 + 3*z^2 + 4q^{8} + 5*z^9 + 2*z^8 + 2*z^7 + 8*z^6 + 8*z^5 + 6*z^4 + 5*z^3 + 4*z^2 + 10*z + 9q^{9} + 6*z^9 + 3*z^8 + 5*z^7 + 8*z^6 + 6*z^5 + 2*z^4 + 7*z^3 + z^2 + 6*z + 1q^{10} + 7*z^9 + 10*z^8 + 10*z^7 + 8*z^5 + 8*z^4 + 10*z^2 + 4*z + 3q^{11} + 2*z^9 + 5*z^8 + 7*z^7 + 7*z^6 + 9*z^5 + 8*z^4 + 5*z^3 + 4*z + 5q^{12} + 5q^{13} + z^9 + 3*z^8 + 6*z^7 + 8*z^6 + 6*z^5 + z^4 + 5*z^3 + 3*z^2 + 10*z + 10q^{14} + 7*z^9 + 6*z^8 + 2*z^7 + 4*z^6 + 9*z^5 + z^4 + 6*z^3 + 5*z^2 + 8*z + 9q^{15} + 8*z^9 + 4*z^8 + 3*z^7 + 10*z^6 + 5*z^5 + 9*z^4 + 7*z^3 + 8*z^2 + 6*z + 3q^{16} + 6*z^9 + 10*z^8 + 3*z^7 + 7*z^6 + 8*z^4 + 10*z^3 + 8*z^2 + 2*z + 10q^{17} + 5*z^9 + 9*z^8 + 7*z^7 + 5*z^6 + z^5 + 10*z^4 + 6*z^3 + 5*z^2 + 10*z + 9q^{18} + 10*z^7 + 3*z^6 + 10*z^5 + 2*z^4 + 9*z^3 + 2*z^2 + 6*z + 10q^{19} + 5*z^9 + z^8 + 9*z^7 + 2*z^5 + 5*z^4 + 5*z^3 + 9q^{20} + 3*z^9 + 3*z^8 + 7*z^7 + z^6 + 8*z^5 + z^4 + z^3 + z^2 + 7*zq^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5*z^9 + 10*z^8 + 8*z^7 + 8*z^6 + 7*z^5 + 7*z^4 + 6*z^3 + z^2 + 7*zq^{2} + z^9 + 10*z^8 + 2*z^7 + 8*z^6 + 3*z^5 + 9*z^4 + 3*z^3 + 9*z^2 + 3*zq^{3} + 4*z^9 + 8*z^8 + 5*z^7 + z^6 + 5*z^5 + 5*z^4 + 8*z^3 + 5*z^2 + 5*z + 8q^{4} + 8*z^9 + 5*z^8 + 9*z^7 + 9*z^6 + 5*z^5 + 7*z^4 + 6*z^3 + 5*z^2 + 6*z + 7q^{5} + 7*z^9 + z^8 + 5*z^7 + 2*z^6 + 5*z^5 + 6*z^4 + 3*z^3 + 2*z^2 + 9*z + 8q^{6} + z^9 + 4*z^8 + 4*z^7 + 3*z^6 + 4*z^5 + 6*z^4 + 10*z^2 + 7*z + 2q^{7} + 2*z^9 + 2*z^7 + 5*z^6 + 10*z^5 + 4*z^4 + z^3 + 3*z + 6q^{8} + z^9 + 3*z^8 + 3*z^7 + 9*z^6 + 6*z^4 + 4*z^3 + z^2 + 4*z + 9q^{9} + 7*z^9 + z^8 + 5*z^7 + 10*z^6 + 8*z^5 + 7*z^4 + 9*z^3 + 10*z^2 + 9q^{10} + 7*z^9 + 6*z^8 + 9*z^7 + 6*z^6 + 4*z^5 + 5*z^4 + 7*z^3 + 3*z^2 + 2*z + 7q^{11} + 9*z^9 + 5*z^8 + 5*z^7 + 5*z^6 + 10*z^5 + 6*z^4 + z^3 + 10*z^2 + 6*z + 4q^{12} + 5q^{13} + 5*z^9 + z^8 + 3*z^7 + 7*z^6 + 7*z^5 + 6*z^4 + 4*z^3 + z^2 + 3*z + 3q^{14} + 2*z^9 + 7*z^8 + 7*z^6 + z^5 + 9*z^4 + 9*z^3 + 10*z^2 + 8*z + 1q^{15} + 8*z^9 + 4*z^8 + 7*z^7 + 3*z^6 + 4*z^5 + 3*z^4 + 5*z^3 + 5*z^2 + 8*z + 6q^{16} + z^9 + 3*z^8 + 6*z^7 + 5*z^6 + 9*z^5 + 2*z^4 + zq^{17} + 7*z^9 + 6*z^6 + 4*z^4 + 5*z^3 + 9*z^2 + 10*z + 7q^{18} + 10*z^9 + 4*z^8 + 8*z^6 + 10*z^5 + z^4 + 9*z^3 + 7*z^2 + 6*z + 2q^{19} + 9*z^9 + 8*z^8 + 10*z^6 + 4*z^5 + 4*z^4 + 8*z^3 + 8*z^2 + 5*z + 10q^{20} + 6*z^9 + 3*z^7 + 2*z^6 + 5*z^5 + 7*z^4 + 8*z^3 + 6*z^2 + 5*zq^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8*z^9 + 8*z^8 + 5*z^7 + 4*z^6 + 7*z^5 + 9*z^4 + 7*z^3 + 5*z^2 + 6q^{2} + 6*z^9 + 9*z^8 + 3*z^7 + 8*z^6 + 10*z^5 + 3*z^4 + 9*z^3 + 6*z^2 + 9*z + 2q^{3} + 6*z^9 + 3*z^8 + z^7 + 4*z^6 + 7*z^5 + 8*z^4 + 9*z^3 + 4*z^2 + 4*z + 8q^{4} + 10*z^9 + 4*z^8 + 6*z^7 + 6*z^5 + 4*z^4 + z^3 + 5*z^2 + 4q^{5} + 7*z^9 + 10*z^8 + 7*z^7 + 6*z^6 + 8*z^5 + 4*z^4 + 2*z^2 + 2*z + 8q^{6} + 10*z^9 + 6*z^8 + 6*z^7 + 7*z^6 + 10*z^5 + 7*z^3 + 7*z^2 + 2*zq^{7} + 8*z^9 + 9*z^8 + 3*z^7 + 2*z^6 + 2*z^5 + 2*z^4 + 7*z^3 + 4*z^2 + 6*zq^{8} + 3*z^8 + 2*z^7 + 2*z^6 + 5*z^5 + 7*z^3 + 2*z^2 + 8q^{9} + 4*z^9 + 8*z^8 + 4*z^7 + 2*z^6 + 6*z^5 + 4*z^4 + 10*z^3 + 5*z^2 + 5*z + 7q^{10} + 5*z^9 + 8*z^8 + 3*z^7 + 4*z^6 + 10*z^5 + 3*z^4 + 3*z^3 + 8*z^2 + 10*z + 6q^{11} + z^9 + 2*z^8 + 8*z^7 + z^6 + 9*z^5 + z^3 + 6*z^2 + 10*z + 3q^{12} + 5q^{13} + 9*z^9 + 4*z^8 + 3*z^7 + 10*z^6 + 9*z^5 + 2*z^4 + 10*z^2 + 8*z + 1q^{14} + 8*z^9 + 3*z^8 + 4*z^7 + 2*z^6 + 4*z^4 + 9*z^3 + 9*z^2 + 7q^{15} + z^9 + z^8 + 9*z^7 + 2*z^6 + 6*z^5 + 3*z^4 + 7*z^3 + 5*z^2 + 9*z + 3q^{16} + z^9 + 2*z^8 + 7*z^7 + z^6 + 3*z^4 + 5*z^3 + 5*z^2 + 7*z + 1q^{17} + 3*z^9 + z^8 + 6*z^6 + 6*z^5 + 10*z^3 + 4*z^2 + 9*z + 9q^{18} + 5*z^9 + 5*z^8 + 7*z^7 + 2*z^6 + 4*z^5 + 7*z^4 + 5*z^3 + z^2 + 6*z + 1q^{19} + 4*z^9 + 6*z^8 + 8*z^7 + 7*z^6 + 4*z^5 + 3*z^4 + 8*z^3 + 8*z^2 + 7*z + 4q^{20} + 6*z^9 + 2*z^8 + 9*z^7 + 10*z^6 + 7*z^5 + 3*z^4 + 10*z^3 + 4*z^2 + 6*z + 9q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8*z^9 + 10*z^8 + 6*z^7 + 7*z^6 + 2*z^5 + 7*z^4 + 6*z^3 + 4*z^2 + 4*z + 3q^{2} + 6*z^9 + 9*z^8 + 7*z^7 + 8*z^6 + 2*z^5 + 2*z^4 + 4*z^3 + 10*z^2 + 10*z + 2q^{3} + 8*z^9 + 5*z^8 + 10*z^7 + 5*z^6 + 7*z^5 + 9*z^4 + 10*z^3 + 2*z^2 + 8*z + 6q^{4} + 2*z^9 + 5*z^8 + z^7 + z^6 + 5*z^5 + 6*z^4 + 8*z^3 + 10*z^2 + z + 10q^{5} + 6*z^9 + 7*z^8 + 2*z^7 + 7*z^6 + 2*z^5 + 8*z^4 + 6*z^3 + 9*z^2 + 3q^{6} + z^9 + 6*z^8 + z^7 + 7*z^6 + 6*z^5 + z^4 + 2*z^3 + 6*z^2 + 2*z + 8q^{7} + 6*z^9 + 5*z^8 + 9*z^7 + 8*z^6 + 9*z^5 + 3*z^4 + 10*z + 7q^{8} + 8*z^9 + 2*z^8 + 8*z^7 + 8*z^6 + 8*z^5 + 4*z^3 + 5*z^2 + z + 10q^{9} + 7*z^9 + 2*z^8 + 9*z^7 + 3*z^6 + 7*z^5 + 2*z^4 + 6*z^3 + 10*z^2 + 9*z + 8q^{10} + 3*z^9 + z^8 + 6*z^7 + 7*z^6 + 4*z^5 + 6*z^3 + 8*z + 9q^{11} + 2*z^9 + 5*z^8 + 8*z^5 + 2*z^4 + 3*z^3 + 8*z^2 + 6*z + 8q^{12} + 5q^{13} + 4*z^9 + 4*z^8 + 10*z^7 + 5*z^6 + 2*z^5 + 3*z^4 + 6*z^3 + 2*z^2 + 4*z + 2q^{14} + 6*z^9 + 4*z^8 + 2*z^7 + z^6 + 8*z^5 + 7*z^4 + 10*z^3 + 5*z^2 + 5*z + 8q^{15} + 5*z^9 + z^8 + 4*z^7 + 8*z^6 + 4*z^5 + 10*z^4 + 9*z^3 + 3*z^2 + 9*z + 7q^{16} + 4*z^9 + 5*z^8 + 7*z^7 + 10*z^6 + 3*z^5 + 4*z^4 + 7*z^3 + 7*z^2 + 6*z + 5q^{17} + 10*z^8 + 3*z^7 + 9*z^6 + 7*z^5 + 9*z^4 + 4*z^3 + 9*z^2 + 7*z + 10q^{18} + 5*z^9 + 10*z^8 + 5*z^6 + 4*z^5 + 10*z^4 + z^2 + 7*z + 8q^{19} + z^9 + 10*z^8 + 3*z^7 + 2*z^6 + 7*z^5 + 8*z^4 + 4*z^2 + 10*z + 8q^{20} + 2*z^9 + 2*z^8 + 9*z^7 + 3*z^6 + 9*z^5 + 4*z^4 + 8*z^3 + 10*z + 7q^{21} + \cdots \in S_{20}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{3} + 9q^{4} + 10q^{5} + 9q^{7} + 9q^{9} + 10q^{11} + 2q^{12} + 10q^{13} + q^{15} + 4q^{16} + 7q^{17} + 2q^{19} + 2q^{20} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4*z^19 + 5*z^18 + 3*z^17 + 7*z^16 + 7*z^15 + 6*z^14 + 6*z^13 + 7*z^12 + z^11 + 4*z^10 + 4*z^9 + 9*z^8 + 8*z^7 + 7*z^5 + 5*z^4 + z^3 + 6*z^2 + 9*z + 6q^{2} + 8*z^19 + 9*z^18 + z^17 + 9*z^16 + 6*z^15 + 2*z^14 + 9*z^13 + 3*z^10 + 4*z^9 + 5*z^8 + 4*z^7 + z^6 + 3*z^4 + 5*z^3 + 4*z^2 + 8*z + 4q^{3} + 3*z^19 + 2*z^18 + 4*z^17 + z^16 + 7*z^15 + 5*z^14 + 6*z^12 + 6*z^11 + 3*z^10 + 8*z^9 + 8*z^8 + 2*z^7 + 10*z^6 + 5*z^5 + 5*z^4 + 10*z^3 + 3*z^2 + 3*z + 2q^{4} + 2*z^19 + 6*z^18 + 9*z^17 + z^16 + 2*z^14 + z^13 + 2*z^12 + z^11 + 2*z^10 + 3*z^9 + 2*z^8 + z^7 + 2*z^6 + 4*z^5 + 4*z^3 + 6*z^2 + z + 8q^{5} + 5*z^19 + 8*z^18 + 2*z^17 + 2*z^16 + 7*z^15 + 7*z^14 + z^13 + 8*z^12 + 7*z^11 + 5*z^10 + 10*z^8 + 3*z^7 + z^6 + 9*z^5 + 5*z^4 + 3*z^3 + 9*z^2 + 4*z + 9q^{6} + 4*z^19 + 8*z^18 + 6*z^17 + 9*z^16 + 2*z^15 + 6*z^14 + 6*z^12 + 8*z^11 + 9*z^10 + 7*z^9 + 8*z^8 + z^7 + 8*z^6 + 4*z^5 + 9*z^4 + 6*z + 1q^{7} + 7*z^19 + 10*z^18 + 10*z^17 + 10*z^16 + 9*z^15 + z^12 + 3*z^11 + z^10 + 4*z^9 + 5*z^8 + 3*z^7 + 7*z^6 + 9*z^5 + 3*z^4 + 10*z^3 + 3*z^2 + 9*z + 3q^{8} + 6*z^19 + 3*z^18 + 4*z^17 + 10*z^16 + 2*z^15 + 8*z^14 + z^13 + 8*z^12 + 9*z^11 + 10*z^9 + 10*z^8 + 2*z^7 + 10*z^6 + 8*z^5 + 9*z^4 + 4*z^3 + 6*z^2 + 7*z + 8q^{9} + 9*z^19 + 8*z^18 + 9*z^17 + 9*z^16 + z^15 + 8*z^14 + 9*z^13 + 4*z^12 + 2*z^11 + 10*z^10 + 3*z^9 + 3*z^8 + 10*z^7 + 6*z^6 + 8*z^5 + 6*z^4 + 7*z^3 + 9*z^2 + 7*zq^{10} + q^{11} + 4*z^19 + 5*z^18 + 9*z^17 + 8*z^16 + 5*z^15 + 10*z^14 + 8*z^13 + z^12 + 6*z^11 + 9*z^10 + 3*z^9 + 7*z^8 + 2*z^7 + 2*z^5 + 8*z^4 + 8*z^3 + 10*z^2 + 9*z + 1q^{12} + 10q^{13} + 3*z^19 + 6*z^18 + z^17 + 8*z^16 + 10*z^15 + 4*z^14 + 8*z^13 + 8*z^12 + 4*z^11 + 2*z^10 + 4*z^9 + 9*z^8 + 7*z^7 + 6*z^6 + 5*z^5 + 5*z^4 + 6*z^3 + 5*z^2 + 10*z + 1q^{14} + 2*z^19 + 6*z^18 + 8*z^17 + 10*z^16 + 4*z^15 + 5*z^14 + 8*z^13 + 3*z^12 + 2*z^11 + 3*z^10 + 5*z^9 + 6*z^8 + 2*z^7 + 2*z^6 + 3*z^5 + 5*z^4 + z^3 + 9*z^2 + z + 5q^{15} + z^19 + 9*z^17 + 8*z^16 + 8*z^15 + 6*z^14 + 9*z^13 + 8*z^11 + 5*z^10 + 5*z^8 + 4*z^6 + 7*z^5 + 8*z^4 + 9*z^3 + 3*z^2 + 9*z + 1q^{16} + z^19 + 2*z^17 + 5*z^16 + 3*z^15 + 5*z^14 + 3*z^13 + 7*z^12 + 4*z^11 + z^10 + 3*z^9 + 7*z^7 + 4*z^6 + 10*z^4 + 10*z^3 + 2*z^2 + 9*z + 6q^{17} + 8*z^19 + 9*z^18 + 7*z^17 + 10*z^16 + 4*z^15 + 6*z^14 + 5*z^12 + 5*z^11 + 8*z^10 + 3*z^9 + 3*z^8 + 9*z^7 + z^6 + 6*z^5 + 6*z^4 + z^3 + 8*z^2 + 8*z + 6q^{18} + 4*z^19 + z^17 + 6*z^16 + 7*z^15 + 8*z^14 + 6*z^13 + 2*z^12 + z^11 + 4*z^9 + 6*z^8 + 2*z^7 + 5*z^6 + 4*z^5 + 9*z^4 + 8*z^3 + 7*z^2 + 3*zq^{19} + 5*z^19 + 10*z^18 + 9*z^17 + 6*z^16 + 2*z^15 + 3*z^14 + 6*z^13 + 6*z^12 + 3*z^11 + 7*z^10 + 3*z^9 + 4*z^8 + 8*z^7 + 10*z^6 + z^5 + z^4 + 10*z^3 + z^2 + 2*z + 1q^{20} + 6*z^19 + 2*z^18 + 9*z^17 + 3*z^16 + 9*z^15 + z^14 + 5*z^13 + 6*z^12 + 8*z^11 + 7*z^10 + 8*z^9 + z^8 + 2*z^7 + 4*z^5 + 7*z^4 + 4*z^3 + z^2 + 8*zq^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10*z^19 + 6*z^18 + 4*z^17 + 10*z^16 + 9*z^15 + 6*z^14 + 7*z^13 + 2*z^12 + 10*z^11 + 9*z^10 + 3*z^9 + 9*z^8 + z^7 + z^6 + 5*z^5 + 9*z^4 + 10*z^3 + 8*z^2 + 7*z + 4q^{2} + 2*z^18 + 3*z^17 + 7*z^16 + 8*z^14 + 2*z^12 + 6*z^10 + 10*z^8 + 9*z^7 + 9*z^6 + 10*z^5 + 5*z^4 + 6*z^3 + 4*z^2 + 9*z + 4q^{3} + 4*z^19 + 9*z^18 + 5*z^17 + 2*z^16 + z^15 + z^14 + z^13 + 4*z^12 + 5*z^11 + 9*z^10 + z^9 + z^8 + 8*z^7 + 7*z^6 + z^5 + 2*z^4 + z^3 + 4*z^2 + 5*z + 1q^{4} + z^19 + 5*z^18 + 9*z^17 + 5*z^16 + 5*z^15 + 10*z^14 + z^13 + 7*z^12 + 10*z^11 + 4*z^9 + 5*z^8 + 8*z^7 + 10*z^6 + 8*z^5 + 3*z^4 + 7*z^3 + 8*z^2 + 4*z + 6q^{5} + 5*z^19 + 3*z^18 + 3*z^17 + 7*z^16 + 6*z^15 + 2*z^13 + 4*z^11 + 9*z^10 + 5*z^9 + 6*z^8 + 5*z^7 + 6*z^6 + 9*z^5 + 5*z^4 + 8*z^3 + z^2 + 9*z + 6q^{6} + z^19 + 3*z^18 + 2*z^17 + z^16 + 10*z^15 + 3*z^14 + 7*z^13 + 9*z^12 + 3*z^11 + 9*z^10 + z^9 + 3*z^7 + z^6 + 5*z^5 + 7*z^4 + 5*z^2 + 6*z + 7q^{7} + 5*z^19 + z^18 + 7*z^17 + 3*z^16 + 4*z^14 + 8*z^13 + 2*z^12 + 8*z^11 + 7*z^10 + 2*z^9 + z^8 + 8*z^6 + 6*z^5 + 9*z^4 + z^3 + 9*z^2 + 8q^{8} + 2*z^19 + 8*z^18 + 6*z^16 + 4*z^15 + 2*z^14 + 8*z^13 + 5*z^12 + 2*z^11 + 9*z^10 + 5*z^9 + 5*z^8 + 2*z^5 + 10*z^4 + 7*z^3 + 2*z^2 + 10*z + 1q^{9} + 2*z^19 + 3*z^18 + 2*z^17 + 2*z^16 + 10*z^15 + 3*z^14 + 2*z^13 + 7*z^12 + 9*z^11 + z^10 + 8*z^9 + 8*z^8 + z^7 + 5*z^6 + 3*z^5 + 5*z^4 + 4*z^3 + 2*z^2 + 4*z + 7q^{10} + q^{11} + 6*z^19 + 6*z^18 + 7*z^17 + z^16 + 8*z^15 + 8*z^14 + 6*z^13 + 7*z^12 + 5*z^11 + 5*z^10 + 2*z^9 + 9*z^8 + 6*z^7 + 7*z^6 + 5*z^5 + 2*z^4 + 3*z^3 + 4*zq^{12} + 10q^{13} + 4*z^19 + 5*z^18 + 8*z^17 + 6*z^16 + 9*z^15 + 2*z^14 + 4*z^13 + 2*z^12 + 7*z^11 + 10*z^10 + 5*z^9 + 3*z^7 + z^5 + 2*z^4 + 5*z^3 + 2*z^2 + 9*z + 4q^{14} + 9*z^19 + 5*z^18 + 3*z^17 + z^16 + 7*z^15 + 6*z^14 + 3*z^13 + 8*z^12 + 9*z^11 + 8*z^10 + 6*z^9 + 5*z^8 + 9*z^7 + 9*z^6 + 8*z^5 + 6*z^4 + 10*z^3 + 2*z^2 + 10*z + 1q^{15} + 10*z^19 + 2*z^17 + 3*z^16 + 3*z^15 + 5*z^14 + 2*z^13 + 3*z^11 + 6*z^10 + 6*z^8 + 7*z^6 + 4*z^5 + 3*z^4 + 2*z^3 + 8*z^2 + 2*z + 7q^{16} + 4*z^17 + 8*z^16 + 6*z^15 + 10*z^14 + 5*z^13 + 7*z^12 + 7*z^11 + 7*z^10 + 3*z^9 + 6*z^8 + 7*z^7 + 4*z^5 + 2*z^4 + z^3 + 10*z^2 + 9q^{17} + 7*z^19 + 2*z^18 + 6*z^17 + 9*z^16 + 10*z^15 + 10*z^14 + 10*z^13 + 7*z^12 + 6*z^11 + 2*z^10 + 10*z^9 + 10*z^8 + 3*z^7 + 4*z^6 + 10*z^5 + 9*z^4 + 10*z^3 + 7*z^2 + 6*z + 7q^{18} + z^19 + 7*z^17 + 4*z^16 + 5*z^15 + z^14 + z^13 + 2*z^12 + 10*z^11 + 7*z^10 + 4*z^9 + 2*z^8 + 2*z^7 + 4*z^6 + 5*z^5 + 7*z^4 + 3*z^3 + 9*z^2 + 9*z + 9q^{19} + 3*z^19 + z^18 + 6*z^17 + 10*z^16 + 4*z^15 + 7*z^14 + 3*z^13 + 7*z^12 + 8*z^11 + 2*z^10 + z^9 + 5*z^7 + 9*z^5 + 7*z^4 + z^3 + 7*z^2 + 4*z + 6q^{20} + z^19 + 9*z^18 + 10*z^15 + 5*z^14 + 7*z^13 + 4*z^12 + 3*z^11 + 5*z^10 + z^9 + 8*z^8 + 8*z^7 + 6*z^6 + 2*z^5 + 7*z^3 + 6*z^2 + 6q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^19 + 6*z^18 + 9*z^17 + 2*z^14 + 6*z^13 + 4*z^12 + 6*z^11 + 10*z^10 + 7*z^9 + 5*z^8 + 3*z^7 + 9*z^6 + 6*z^5 + 2*z^4 + 9*z^2 + 3*z + 8q^{2} + 6*z^19 + 3*z^18 + 6*z^17 + 3*z^16 + 5*z^15 + 2*z^14 + 9*z^13 + 6*z^12 + 7*z^11 + 9*z^10 + 6*z^9 + 2*z^8 + 10*z^6 + 10*z^5 + 10*z^4 + 10*z^3 + 7*z + 3q^{3} + 3*z^19 + z^18 + z^17 + 2*z^15 + 5*z^13 + z^12 + 2*z^11 + 7*z^10 + 9*z^8 + 10*z^7 + 4*z^5 + 10*z^4 + 8*z^3 + 4*z + 2q^{4} + 7*z^19 + 10*z^18 + 5*z^17 + 7*z^15 + 4*z^14 + 2*z^13 + 6*z^12 + 8*z^11 + 6*z^10 + 3*z^9 + 3*z^8 + 8*z^7 + 7*z^6 + 4*z^5 + 6*z^4 + 6*z^3 + 7*z^2 + 9*z + 1q^{5} + 10*z^19 + 6*z^17 + 9*z^15 + 4*z^14 + 7*z^13 + 7*z^12 + 10*z^11 + 2*z^10 + 3*z^9 + z^8 + 7*z^7 + 7*z^6 + 8*z^5 + 5*z^4 + 3*z^3 + 7*z^2 + 2*z + 2q^{6} + 7*z^19 + 8*z^18 + 4*z^17 + 8*z^16 + 10*z^15 + z^14 + 10*z^13 + 2*z^12 + 6*z^11 + 9*z^10 + 10*z^9 + 7*z^8 + 8*z^7 + 9*z^6 + 7*z^5 + 2*z^4 + 6*z^3 + 8*z^2 + 5q^{7} + 10*z^19 + 9*z^18 + 5*z^17 + 8*z^16 + z^15 + z^14 + 4*z^13 + 3*z^12 + 8*z^11 + 5*z^10 + 10*z^9 + 5*z^8 + 7*z^7 + 9*z^6 + z^4 + 3*z^3 + 8*z^2 + 4*z + 7q^{8} + 3*z^19 + 7*z^18 + 9*z^17 + 8*z^16 + 6*z^15 + 5*z^14 + z^13 + 8*z^12 + 3*z^11 + 4*z^10 + 2*z^9 + 10*z^8 + 5*z^7 + 5*z^6 + 8*z^4 + z^3 + 4*z^2 + 9*z + 5q^{9} + 4*z^19 + 4*z^18 + 6*z^16 + 2*z^15 + z^14 + 2*z^12 + 8*z^11 + 8*z^10 + 7*z^9 + 10*z^8 + 5*z^7 + z^6 + 6*z^5 + 10*z^4 + 10*z^3 + 3*z^2 + 10*z + 6q^{10} + q^{11} + 3*z^19 + 2*z^18 + 2*z^17 + 8*z^16 + 4*z^15 + 3*z^13 + 8*z^12 + 8*z^11 + 5*z^10 + z^9 + 5*z^8 + 4*z^7 + 10*z^6 + 3*z^5 + 4*z^4 + 4*z^3 + 9*z^2 + 3*z + 4q^{12} + 10q^{13} + 5*z^19 + z^18 + 8*z^17 + 5*z^16 + 7*z^15 + z^14 + z^13 + z^12 + 7*z^11 + 6*z^10 + 8*z^8 + 10*z^7 + 8*z^6 + 7*z^5 + 4*z^4 + 4*z^3 + 6*z^2 + 8q^{14} + 3*z^19 + 7*z^18 + 8*z^17 + 6*z^16 + 10*z^15 + 8*z^14 + 8*z^13 + 9*z^12 + 4*z^11 + 5*z^10 + 4*z^9 + 3*z^8 + 6*z^7 + 5*z^6 + 10*z^5 + 2*z^4 + 9*z^3 + 7*z^2 + 9*z + 7q^{15} + 9*z^18 + 7*z^17 + 2*z^16 + 4*z^15 + 6*z^14 + 7*z^13 + 10*z^12 + z^11 + 6*z^10 + 2*z^9 + 5*z^8 + 3*z^7 + 2*z^6 + 2*z^3 + 10*z^2 + 2*z + 5q^{16} + 5*z^19 + 10*z^18 + 6*z^17 + 3*z^15 + z^14 + 5*z^13 + 9*z^12 + 6*z^11 + 10*z^10 + 9*z^9 + 5*z^8 + 10*z^6 + 9*z^5 + 5*z^4 + z^3 + 10*z^2 + 2*z + 7q^{17} + 8*z^19 + 10*z^18 + 10*z^17 + 9*z^15 + 6*z^13 + 10*z^12 + 9*z^11 + 4*z^10 + 2*z^8 + z^7 + 7*z^5 + z^4 + 3*z^3 + 7*z + 6q^{18} + 3*z^19 + 8*z^18 + z^17 + 9*z^16 + 10*z^14 + 7*z^13 + 2*z^12 + 7*z^11 + 10*z^9 + 9*z^8 + 8*z^7 + 4*z^6 + z^5 + 3*z^4 + 3*z^3 + 7*z^2 + 8*z + 1q^{19} + z^19 + 9*z^18 + 6*z^17 + z^16 + 8*z^15 + 9*z^14 + 9*z^13 + 9*z^12 + 8*z^11 + 10*z^10 + 6*z^8 + 2*z^7 + 6*z^6 + 8*z^5 + 3*z^4 + 3*z^3 + 10*z^2 + 9q^{20} + 3*z^19 + 6*z^18 + 6*z^16 + 3*z^15 + 7*z^14 + 4*z^13 + 9*z^12 + 5*z^11 + 3*z^10 + 6*z^9 + 2*z^8 + 8*z^7 + 6*z^6 + 4*z^5 + 10*z^4 + 3*z^3 + 8*z^2 + 10*z + 1q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7*z^19 + 5*z^18 + 6*z^17 + 5*z^16 + 6*z^15 + 8*z^14 + 3*z^13 + 9*z^12 + 5*z^11 + 10*z^10 + 8*z^9 + 10*z^8 + 10*z^7 + z^6 + 4*z^5 + 6*z^4 + 10*z^2 + 3*z + 7q^{2} + 8*z^19 + 8*z^18 + z^17 + 3*z^16 + 10*z^14 + 4*z^13 + 3*z^12 + 4*z^11 + 4*z^10 + z^9 + 5*z^8 + 9*z^7 + 2*z^6 + 2*z^5 + 4*z^4 + z^3 + 3*z^2 + 9*zq^{3} + z^19 + 10*z^18 + z^17 + 8*z^16 + z^15 + 5*z^14 + 5*z^13 + 9*z^11 + 3*z^10 + 2*z^9 + 4*z^8 + 2*z^7 + 5*z^6 + z^5 + 5*z^4 + 3*z^3 + 4*z^2 + 10*z + 9q^{4} + z^19 + z^18 + 10*z^17 + 5*z^16 + 10*z^15 + 6*z^14 + 7*z^13 + 7*z^12 + 3*z^11 + 3*z^10 + z^9 + z^8 + 5*z^7 + 3*z^6 + 6*z^5 + 2*z^4 + 5*z^3 + z^2 + 8*z + 7q^{5} + 2*z^19 + 2*z^16 + z^13 + 7*z^12 + z^11 + 6*z^10 + 3*z^9 + 5*z^8 + 7*z^7 + 8*z^6 + 7*z^5 + 7*z^4 + 8*z^3 + 5*z^2 + 7*z + 4q^{6} + 10*z^19 + 3*z^18 + 10*z^17 + 4*z^16 + z^14 + 5*z^13 + 5*z^12 + 5*z^11 + 6*z^10 + 4*z^9 + 7*z^8 + 10*z^7 + 4*z^6 + 6*z^5 + 4*z^4 + 5*z^3 + 9*z^2 + 10*z + 4q^{7} + 2*z^18 + z^16 + z^15 + 6*z^14 + 10*z^13 + 5*z^12 + 3*z^11 + 9*z^10 + 6*z^9 + z^7 + 9*z^6 + 7*z^5 + 9*z^4 + 8*z^3 + 2*z^2 + 9*z + 2q^{8} + 4*z^18 + 9*z^17 + 9*z^16 + 10*z^15 + 7*z^14 + z^13 + z^12 + 8*z^11 + 9*z^10 + 5*z^9 + 8*z^8 + 4*z^7 + 7*z^6 + z^5 + 6*z^4 + 10*z^3 + 10*z^2 + 7*z + 10q^{9} + 7*z^19 + 7*z^18 + 5*z^16 + 9*z^15 + 10*z^14 + 9*z^12 + 3*z^11 + 3*z^10 + 4*z^9 + z^8 + 6*z^7 + 10*z^6 + 5*z^5 + z^4 + z^3 + 8*z^2 + z + 1q^{10} + q^{11} + 9*z^19 + 9*z^18 + 4*z^17 + 5*z^16 + 5*z^15 + 4*z^14 + 5*z^13 + 6*z^12 + 3*z^11 + 3*z^10 + 5*z^9 + z^8 + 10*z^7 + 5*z^6 + z^5 + 8*z^4 + 7*z^3 + 3*z^2 + 6*z + 10q^{12} + 10q^{13} + 10*z^19 + 10*z^18 + 5*z^17 + 3*z^16 + 7*z^15 + 4*z^14 + 9*z^13 + 4*z^11 + 4*z^10 + 2*z^9 + 5*z^8 + 2*z^7 + 8*z^6 + 9*z^5 + 7*z^3 + 9*z^2 + 3*z + 5q^{14} + 8*z^19 + 4*z^18 + 3*z^17 + 5*z^16 + z^15 + 3*z^14 + 3*z^13 + 2*z^12 + 7*z^11 + 6*z^10 + 7*z^9 + 8*z^8 + 5*z^7 + 6*z^6 + z^5 + 9*z^4 + 2*z^3 + 4*z^2 + 2*z + 10q^{15} + 2*z^18 + 4*z^17 + 9*z^16 + 7*z^15 + 5*z^14 + 4*z^13 + z^12 + 10*z^11 + 5*z^10 + 9*z^9 + 6*z^8 + 8*z^7 + 9*z^6 + 9*z^3 + z^2 + 9*z + 3q^{16} + 5*z^19 + z^18 + 10*z^17 + 9*z^16 + 10*z^15 + 6*z^14 + 9*z^13 + 10*z^12 + 5*z^11 + 4*z^10 + 7*z^9 + 8*z^7 + 8*z^6 + 9*z^5 + 5*z^4 + 10*z^3 + 6q^{17} + 10*z^19 + z^18 + 10*z^17 + 3*z^16 + 10*z^15 + 6*z^14 + 6*z^13 + 2*z^11 + 8*z^10 + 9*z^9 + 7*z^8 + 9*z^7 + 6*z^6 + 10*z^5 + 6*z^4 + 8*z^3 + 7*z^2 + z + 10q^{18} + 3*z^19 + 3*z^18 + 2*z^17 + 3*z^16 + 10*z^15 + 3*z^14 + 8*z^13 + 5*z^12 + 4*z^11 + 4*z^10 + 4*z^9 + 5*z^8 + 10*z^7 + 9*z^6 + z^5 + 3*z^4 + 8*z^3 + 10*z^2 + 2*z + 9q^{19} + 2*z^19 + 2*z^18 + z^17 + 5*z^16 + 8*z^15 + 3*z^14 + 4*z^13 + 3*z^11 + 3*z^10 + 7*z^9 + z^8 + 7*z^7 + 6*z^6 + 4*z^5 + 8*z^3 + 4*z^2 + 5*z + 4q^{20} + z^19 + 5*z^18 + 2*z^17 + 2*z^16 + 9*z^14 + 6*z^13 + 3*z^12 + 6*z^11 + 7*z^10 + 7*z^9 + 4*z^7 + 10*z^6 + z^5 + 5*z^4 + 8*z^3 + 7*z^2 + 4*z + 2q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{20} + 6x^{19} + 5x^{18} + 9x^{17} + 9x^{16} + 2x^{15} + x^{13} + 5x^{12} + 7x^{11} + 10x^{10} + 7x^{9} + 10x^{8} + 9x^{6} + x^{5} + 6x^{4} + 7x^{3} + x + 4=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^11 + 6*z^10 + 5*z^9 + 5*z^8 + 2*z^7 + 3*z^6 + 3*z^5 + 2*z^4 + 6*z^3 + z^2 + 7q^{2} + 7*z^10 + 7*z^9 + 7*z^8 + 10*z^7 + 7*z^6 + 5*z^4 + 4*z^3 + 7*z^2 + 3*z + 1q^{3} + 6*z^11 + 4*z^10 + 6*z^9 + z^8 + 10*z^6 + 8*z^5 + 4*z^4 + 2*z^2 + 6*z + 8q^{4} + 8*z^11 + 10*z^10 + 5*z^9 + 10*z^8 + 5*z^7 + 6*z^6 + z^5 + 7*z^4 + 8*z^2 + 8*z + 2q^{5} + 8*z^11 + 8*z^10 + 6*z^7 + 6*z^6 + 2*z^5 + z^2 + 8*z + 2q^{6} + 4*z^11 + 10*z^10 + 7*z^9 + 5*z^8 + 4*z^7 + z^6 + 8*z^5 + 3*z^4 + 2*z^3 + 10*z^2 + 3q^{7} + z^11 + 5*z^10 + 6*z^9 + 2*z^8 + 9*z^7 + 8*z^6 + 6*z^5 + 5*z^4 + z^3 + 7*z^2 + 10*zq^{8} + 10*z^11 + 2*z^10 + 10*z^9 + 7*z^8 + 2*z^7 + 4*z^6 + 2*z^5 + z^4 + 9*z^3 + 4*z^2 + 3*z + 9q^{9} + 8*z^10 + 8*z^9 + 8*z^8 + 2*z^7 + 8*z^6 + z^4 + 3*z^3 + 8*z^2 + 5*zq^{10} + 10q^{11} + 9*z^11 + 4*z^8 + 2*z^5 + 4*z^4 + 4*z^3 + 3*z^2 + z + 2q^{12} + q^{13} + 7*z^11 + 7*z^10 + 7*z^9 + 4*z^8 + 10*z^7 + 7*z^6 + 4*z^5 + 2*z^4 + z^3 + 2*z^2 + 5*z + 3q^{14} + 3*z^11 + 8*z^10 + 3*z^9 + 7*z^8 + 10*z^7 + 4*z^6 + 6*z^5 + 3*z^4 + z^3 + 8*z^2 + z + 4q^{15} + 8*z^11 + 8*z^10 + 8*z^9 + 3*z^8 + 2*z^7 + 8*z^6 + 3*z^5 + 7*z^4 + 9*z^3 + 7*z^2 + z + 10q^{16} + 9*z^11 + 10*z^10 + z^9 + z^8 + 7*z^7 + 5*z^6 + 5*z^5 + 7*z^4 + 10*z^3 + 9*z^2 + 8q^{17} + 7*z^11 + 3*z^10 + 3*z^8 + 5*z^7 + 5*z^6 + 5*z^5 + 3*z^4 + 3*z^3 + 4*z^2 + z + 7q^{18} + 3*z^11 + 8*z^10 + 2*z^9 + 8*z^8 + 5*z^7 + z^6 + 10*z^5 + 9*z^4 + 4*z^3 + 10*z^2 + 6*z + 8q^{19} + 3*z^11 + 9*z^10 + z^8 + 4*z^7 + 4*z^6 + z^4 + z^3 + 6*z^2 + z + 6q^{20} + 4*z^11 + 3*z^10 + 8*z^9 + z^8 + z^7 + 7*z^6 + 9*z^5 + 5*z^4 + 7*z^3 + 9*z^2 + z + 8q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4*z^11 + 9*z^10 + 3*z^9 + 7*z^8 + 8*z^7 + 2*z^6 + 9*z^5 + 3*z^4 + z^3 + 4*z^2 + 2*z + 3q^{2} + 10*z^11 + 2*z^10 + 6*z^9 + 4*z^7 + 3*z^6 + 7*z^5 + 3*z^4 + 10*z^3 + z^2 + z + 2q^{3} + 3*z^11 + 7*z^9 + 9*z^8 + 2*z^7 + 10*z^6 + 2*z^5 + 7*z^4 + 6*z^3 + 9*z^2 + 2*z + 7q^{4} + 8*z^11 + 7*z^10 + 8*z^9 + z^8 + 4*z^7 + 10*z^6 + 10*z^5 + 5*z^4 + 7*z^3 + 4*z^2 + 5*z + 3q^{5} + 4*z^10 + 7*z^9 + z^8 + 5*z^7 + 2*z^6 + 10*z^5 + 10*z^4 + 9*z^3 + 9*z^2 + 4*z + 2q^{6} + 4*z^11 + 4*z^10 + 5*z^9 + 6*z^8 + 6*z^7 + 7*z^6 + 3*z^5 + 3*z^4 + 7*z^3 + 7*z^2 + z + 2q^{7} + 6*z^11 + 3*z^10 + 9*z^9 + 7*z^8 + 6*z^7 + 10*z^6 + 3*z^5 + 6*z^4 + 4*z^2 + 6*zq^{8} + 10*z^11 + 9*z^9 + 4*z^8 + z^7 + 5*z^6 + 9*z^5 + 3*z^4 + 8*z^3 + 5*z + 9q^{9} + 2*z^11 + 7*z^10 + 10*z^9 + 3*z^7 + 5*z^6 + 8*z^5 + 5*z^4 + 2*z^3 + 9*z^2 + 9*z + 9q^{10} + 10q^{11} + z^11 + 10*z^10 + 10*z^9 + 8*z^8 + 8*z^7 + 10*z^6 + 10*z^5 + 2*z^4 + 10*z^3 + 3*z^2 + 3*z + 6q^{12} + q^{13} + z^11 + 4*z^9 + 5*z^8 + 9*z^7 + z^6 + 5*z^5 + 7*z^4 + 8*z^3 + 7*z^2 + 7*z + 1q^{14} + 10*z^11 + 3*z^9 + 10*z^8 + 4*z^7 + 9*z^6 + 6*z^4 + 4*z^3 + z^2 + 2*z + 10q^{15} + 9*z^11 + 3*z^9 + z^8 + 4*z^7 + 9*z^6 + z^5 + 8*z^4 + 6*z^3 + 8*z^2 + 8*z + 3q^{16} + 3*z^11 + 4*z^10 + 5*z^9 + 8*z^8 + 6*z^7 + 7*z^6 + 4*z^5 + 5*z^4 + 9*z^3 + 3*z^2 + 7*z + 5q^{17} + 9*z^11 + 9*z^10 + 6*z^9 + 5*z^8 + z^7 + 3*z^5 + 8*z^4 + 4*z^3 + 7*z^2 + z + 10q^{18} + 2*z^11 + 6*z^10 + 9*z^9 + 10*z^8 + 4*z^6 + 8*z^5 + 9*z^4 + 3*z^3 + 8*z^2 + 7*z + 8q^{19} + 3*z^11 + 7*z^10 + 9*z^9 + 10*z^8 + 9*z^7 + 2*z^6 + 9*z^4 + 3*z^3 + 4*z^2 + 8*z + 7q^{20} + 3*z^11 + 9*z^10 + 6*z^9 + 6*z^8 + z^7 + 9*z^5 + 9*z^4 + 5*z^3 + 5*z^2 + 4*z + 10q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 7q^{4} + 6q^{5} + q^{6} + 8q^{7} + 4q^{8} + 2q^{9} + 7q^{10} + 6q^{12} + q^{13} + 2q^{14} + 2q^{15} + 9q^{16} + 7q^{17} + 6q^{18} + 9q^{19} + 9q^{20} + 10q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 7*z^11 + z^10 + 6*z^9 + 4*z^8 + 6*z^7 + 5*z^6 + 6*z^5 + 7*z^4 + 3*z^3 + 8*z^2 + z + 5q^{2} + z^11 + 9*z^9 + z^7 + 5*z^6 + 7*z^5 + 10*z^4 + 4*z^3 + 8*z^2 + 4*z + 4q^{3} + 8*z^11 + 9*z^10 + 2*z^8 + 4*z^7 + 4*z^6 + 6*z^5 + 2*z^4 + 2*z^3 + 4*z^2 + 4*z + 6q^{4} + 3*z^11 + 5*z^10 + 5*z^9 + 10*z^8 + 4*z^7 + 5*z^6 + 8*z^5 + 7*z^4 + 6*z^2 + 3*z + 1q^{5} + 8*z^10 + 6*z^9 + 10*z^8 + 3*z^7 + 2*z^6 + 8*z^5 + 2*z^4 + 9*z^3 + z^2 + 4*z + 5q^{6} + 7*z^11 + 3*z^10 + 9*z^9 + 5*z^8 + 6*z^7 + 10*z^6 + 2*z^5 + 4*z^4 + 9*z^3 + 8*z^2 + 1q^{7} + 5*z^11 + 10*z^10 + 6*z^9 + 4*z^8 + 10*z^7 + 9*z^6 + 7*z^4 + 3*z^3 + 5*z^2 + 10*z + 9q^{8} + z^11 + 3*z^10 + 8*z^9 + 7*z^8 + z^7 + 7*z^6 + z^5 + 2*z^3 + 8*z^2 + 8*z + 1q^{9} + 9*z^11 + 4*z^9 + 9*z^7 + z^6 + 8*z^5 + 2*z^4 + 3*z^3 + 6*z^2 + 3*z + 5q^{10} + 10q^{11} + 10*z^11 + 10*z^9 + 3*z^8 + 6*z^7 + 8*z^6 + 5*z^5 + 8*z^4 + 5*z^3 + 9*z^2 + 6q^{12} + q^{13} + 10*z^11 + 7*z^9 + 6*z^8 + 2*z^7 + 10*z^6 + 6*z^5 + 4*z^4 + 3*z^3 + 4*z^2 + 4*z + 3q^{14} + z^11 + 5*z^10 + 7*z^9 + z^8 + 3*z^7 + 2*z^5 + 10*z^4 + 9*z^3 + 5*z^2 + 5*z + 9q^{15} + 2*z^11 + 8*z^9 + 10*z^8 + 7*z^7 + 2*z^6 + 10*z^5 + 3*z^4 + 5*z^3 + 3*z^2 + 3*z + 10q^{16} + 8*z^11 + 9*z^10 + 10*z^9 + 3*z^8 + 10*z^7 + z^6 + 10*z^5 + 8*z^4 + 5*z^3 + 6*z^2 + 9*z + 1q^{17} + 2*z^11 + 3*z^10 + 7*z^9 + 6*z^8 + 7*z^7 + 4*z^6 + 4*z^5 + 4*z^4 + 3*z^3 + 3*z^2 + 7*z + 7q^{18} + 9*z^11 + 6*z^10 + 4*z^9 + z^8 + 8*z^7 + 10*z^5 + 3*z^4 + 4*z^3 + 2*z^2 + z + 1q^{19} + 8*z^11 + 9*z^10 + z^9 + z^8 + 9*z^7 + 7*z^6 + 2*z^5 + 7*z^4 + 10*z^3 + 2*z^2 + 10*z + 9q^{20} + 8*z^11 + 6*z^10 + 2*z^9 + 5*z^8 + 9*z^7 + 5*z^6 + 8*z^5 + 6*z^4 + z^3 + 2*z^2 + 6*zq^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10*z^11 + 6*z^10 + 8*z^9 + 6*z^8 + 6*z^7 + z^6 + 4*z^5 + 10*z^4 + z^3 + 9*z^2 + 8*z + 9q^{2} + 2*z^10 + 4*z^8 + 7*z^7 + 7*z^6 + 8*z^5 + 4*z^4 + 4*z^3 + 6*z^2 + 3*zq^{3} + 5*z^11 + 9*z^10 + 9*z^9 + 10*z^8 + 5*z^7 + 9*z^6 + 6*z^5 + 9*z^4 + 3*z^3 + 7*z^2 + 10*z + 10q^{4} + 3*z^11 + 4*z^9 + z^8 + 9*z^7 + z^6 + 3*z^5 + 3*z^4 + 4*z^3 + 4*z^2 + 6*z + 5q^{5} + 3*z^11 + 2*z^10 + 9*z^9 + 8*z^7 + z^6 + 2*z^5 + 10*z^4 + 4*z^3 + 6*z + 8q^{6} + 7*z^11 + 5*z^10 + z^9 + 6*z^8 + 6*z^7 + 4*z^6 + 9*z^5 + z^4 + 4*z^3 + 8*z^2 + 10*z + 10q^{7} + 10*z^11 + 4*z^10 + z^9 + 9*z^8 + 8*z^7 + 6*z^6 + 2*z^5 + 4*z^4 + 7*z^3 + 6*z^2 + 7*z + 4q^{8} + z^11 + 6*z^10 + 6*z^9 + 4*z^8 + 7*z^7 + 6*z^6 + 10*z^5 + 7*z^4 + 3*z^3 + 10*z^2 + 6*z + 10q^{9} + 7*z^10 + 3*z^8 + 8*z^7 + 8*z^6 + 6*z^5 + 3*z^4 + 3*z^3 + 10*z^2 + 5*z + 2q^{10} + 10q^{11} + 2*z^11 + z^10 + 2*z^9 + 7*z^8 + 8*z^7 + 4*z^6 + 5*z^5 + 8*z^4 + 3*z^3 + 7*z^2 + 7*z + 7q^{12} + q^{13} + 4*z^11 + 4*z^10 + 4*z^9 + 7*z^8 + z^7 + 4*z^6 + 7*z^5 + 9*z^4 + 10*z^3 + 9*z^2 + 6*z + 1q^{14} + 8*z^11 + 9*z^10 + 9*z^9 + 4*z^8 + 5*z^7 + 9*z^6 + 3*z^5 + 3*z^4 + 8*z^3 + 8*z^2 + 3*z + 8q^{15} + 3*z^11 + 3*z^10 + 3*z^9 + 8*z^8 + 9*z^7 + 3*z^6 + 8*z^5 + 4*z^4 + 2*z^3 + 4*z^2 + 10*z + 3q^{16} + 2*z^11 + 10*z^10 + 6*z^9 + 10*z^8 + 10*z^7 + 9*z^6 + 3*z^5 + 2*z^4 + 9*z^3 + 4*z^2 + 6*z + 4q^{17} + 4*z^11 + 7*z^10 + 9*z^9 + 8*z^8 + 9*z^7 + 2*z^6 + 10*z^5 + 7*z^4 + z^3 + 8*z^2 + 2*z + 2q^{18} + 8*z^11 + 2*z^10 + 7*z^9 + 3*z^8 + 9*z^7 + 6*z^6 + 5*z^5 + z^4 + 2*z^2 + 8*z + 4q^{19} + 8*z^11 + 8*z^10 + z^9 + 10*z^8 + 9*z^6 + 9*z^5 + 5*z^4 + 8*z^3 + 10*z^2 + 3*z + 3q^{20} + 7*z^11 + 4*z^10 + 6*z^9 + 10*z^8 + 10*z^6 + 7*z^5 + 2*z^4 + 9*z^3 + 6*z^2 + 3q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3*z^10 + 6*z^9 + 2*z^7 + z^6 + 10*z^5 + 3*z^4 + 10*z^3 + 8*z^2 + 2*zq^{2} + 5*z^10 + 10*z^9 + 7*z^7 + 9*z^6 + 2*z^5 + 5*z^4 + 2*z^3 + 6*z^2 + 7*z + 7q^{3} + 5*z^10 + 10*z^9 + 7*z^7 + 9*z^6 + 2*z^5 + 5*z^4 + 2*z^3 + 6*z^2 + 7*z + 3q^{4} + 6*z^10 + z^9 + 4*z^7 + 2*z^6 + 9*z^5 + 6*z^4 + 9*z^3 + 5*z^2 + 4*z + 8q^{5} + q^{6} + 3*z^10 + 6*z^9 + 2*z^7 + z^6 + 10*z^5 + 3*z^4 + 10*z^3 + 8*z^2 + 2*z + 6q^{7} + 4*z^10 + 8*z^9 + 10*z^7 + 5*z^6 + 6*z^5 + 4*z^4 + 6*z^3 + 7*z^2 + 10*z + 1q^{8} + 2*z^10 + 4*z^9 + 5*z^7 + 8*z^6 + 3*z^5 + 2*z^4 + 3*z^3 + 9*z^2 + 5*zq^{9} + z^10 + 2*z^9 + 8*z^7 + 4*z^6 + 7*z^5 + z^4 + 7*z^3 + 10*z^2 + 8*z + 10q^{10} + 10q^{11} + 4*z^10 + 8*z^9 + 10*z^7 + 5*z^6 + 6*z^5 + 4*z^4 + 6*z^3 + 7*z^2 + 10*z + 8q^{12} + q^{13} + z^10 + 2*z^9 + 8*z^7 + 4*z^6 + 7*z^5 + z^4 + 7*z^3 + 10*z^2 + 8*z + 5q^{14} + 7*z^10 + 3*z^9 + z^7 + 6*z^6 + 5*z^5 + 7*z^4 + 5*z^3 + 4*z^2 + z + 3q^{15} + 7*z^10 + 3*z^9 + z^7 + 6*z^6 + 5*z^5 + 7*z^4 + 5*z^3 + 4*z^2 + z + 8q^{16} + 5*z^10 + 10*z^9 + 7*z^7 + 9*z^6 + 2*z^5 + 5*z^4 + 2*z^3 + 6*z^2 + 7*zq^{17} + 7*z^10 + 3*z^9 + z^7 + 6*z^6 + 5*z^5 + 7*z^4 + 5*z^3 + 4*z^2 + z + 7q^{18} + 10*z^10 + 9*z^9 + 3*z^7 + 7*z^6 + 4*z^5 + 10*z^4 + 4*z^3 + z^2 + 3*z + 8q^{19} + 5*z^10 + 10*z^9 + 7*z^7 + 9*z^6 + 2*z^5 + 5*z^4 + 2*z^3 + 6*z^2 + 7*z + 4q^{20} + 8*z^10 + 5*z^9 + 9*z^7 + 10*z^6 + z^5 + 8*z^4 + z^3 + 3*z^2 + 9*z + 10q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8*z^10 + 5*z^9 + 9*z^7 + 10*z^6 + z^5 + 8*z^4 + z^3 + 3*z^2 + 9*z + 9q^{2} + 6*z^10 + z^9 + 4*z^7 + 2*z^6 + 9*z^5 + 6*z^4 + 9*z^3 + 5*z^2 + 4*zq^{3} + 6*z^10 + z^9 + 4*z^7 + 2*z^6 + 9*z^5 + 6*z^4 + 9*z^3 + 5*z^2 + 4*z + 7q^{4} + 5*z^10 + 10*z^9 + 7*z^7 + 9*z^6 + 2*z^5 + 5*z^4 + 2*z^3 + 6*z^2 + 7*z + 4q^{5} + q^{6} + 8*z^10 + 5*z^9 + 9*z^7 + 10*z^6 + z^5 + 8*z^4 + z^3 + 3*z^2 + 9*z + 4q^{7} + 7*z^10 + 3*z^9 + z^7 + 6*z^6 + 5*z^5 + 7*z^4 + 5*z^3 + 4*z^2 + z + 2q^{8} + 9*z^10 + 7*z^9 + 6*z^7 + 3*z^6 + 8*z^5 + 9*z^4 + 8*z^3 + 2*z^2 + 6*z + 6q^{9} + 10*z^10 + 9*z^9 + 3*z^7 + 7*z^6 + 4*z^5 + 10*z^4 + 4*z^3 + z^2 + 3*z + 2q^{10} + 10q^{11} + 7*z^10 + 3*z^9 + z^7 + 6*z^6 + 5*z^5 + 7*z^4 + 5*z^3 + 4*z^2 + z + 9q^{12} + q^{13} + 10*z^10 + 9*z^9 + 3*z^7 + 7*z^6 + 4*z^5 + 10*z^4 + 4*z^3 + z^2 + 3*z + 8q^{14} + 4*z^10 + 8*z^9 + 10*z^7 + 5*z^6 + 6*z^5 + 4*z^4 + 6*z^3 + 7*z^2 + 10*z + 2q^{15} + 4*z^10 + 8*z^9 + 10*z^7 + 5*z^6 + 6*z^5 + 4*z^4 + 6*z^3 + 7*z^2 + 10*z + 7q^{16} + 6*z^10 + z^9 + 4*z^7 + 2*z^6 + 9*z^5 + 6*z^4 + 9*z^3 + 5*z^2 + 4*z + 4q^{17} + 4*z^10 + 8*z^9 + 10*z^7 + 5*z^6 + 6*z^5 + 4*z^4 + 6*z^3 + 7*z^2 + 10*z + 6q^{18} + z^10 + 2*z^9 + 8*z^7 + 4*z^6 + 7*z^5 + z^4 + 7*z^3 + 10*z^2 + 8*z + 5q^{19} + 6*z^10 + z^9 + 4*z^7 + 2*z^6 + 9*z^5 + 6*z^4 + 9*z^3 + 5*z^2 + 4*z + 8q^{20} + 3*z^10 + 6*z^9 + 2*z^7 + z^6 + 10*z^5 + 3*z^4 + 10*z^3 + 8*z^2 + 2*z + 1q^{21} + \cdots \in S_{22}(\Gamma_0(13);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 14, \quad \ell = 11}$ \quad (38 forms)}

\begin{enumerate}
\item  Consider
$f = q + 2q^{2} + z^88q^{3} + 4q^{4} + z^25q^{5} + z^100q^{6} + 4q^{7} + 8q^{8} + z^74q^{9} + z^37q^{10} + z^112q^{12} + z^55q^{13} + 8q^{14} + z^113q^{15} + 5q^{16} + z^78q^{17} + z^86q^{18} + z^7q^{19} + z^49q^{20} + z^112q^{21} + \cdots \in S_{14}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{2} + z^16q^{3} + 3q^{4} + z^49q^{5} + z^4q^{6} + 3q^{7} + 7q^{8} + z^50q^{9} + z^37q^{10} + z^91q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{1}&0\\0&u(z^91)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -64q^{2} + aq^{3} + 4096q^{4} + 63*a - 13986q^{5} + -64*aq^{6} + 117649q^{7} + -262144q^{8} + 952*a - 1420383q^{9} + -4032*a + 895104q^{10} + 1008*11*a - 541296*11q^{11} + 4096*aq^{12} + -50895*a + 25981214q^{13} + -7529536q^{14} + 45990*a + 10958220q^{15} + 16777216q^{16} + 128286*a + 47791842q^{17} + -60928*a + 90904512q^{18} + -114129*a + 349993280q^{19} + 258048*a - 57286656q^{20} + 117649*aq^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -952x + -173940$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + z^8q^{3} + 4q^{4} + z^35q^{5} + z^20q^{6} + 4q^{7} + 8q^{8} + z^94q^{9} + z^47q^{10} + z^32q^{12} + z^5q^{13} + 8q^{14} + z^43q^{15} + 5q^{16} + z^18q^{17} + z^106q^{18} + z^77q^{19} + z^59q^{20} + z^32q^{21} + \cdots \in S_{14}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{2} + z^56q^{3} + 3q^{4} + z^59q^{5} + z^44q^{6} + 3q^{7} + 7q^{8} + z^70q^{9} + z^47q^{10} + z^41q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^79)\omega^{1}&0\\0&u(z^41)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -64q^{2} + aq^{3} + 4096q^{4} + 63*a - 13986q^{5} + -64*aq^{6} + 117649q^{7} + -262144q^{8} + 952*a - 1420383q^{9} + -4032*a + 895104q^{10} + 1008*11*a - 541296*11q^{11} + 4096*aq^{12} + -50895*a + 25981214q^{13} + -7529536q^{14} + 45990*a + 10958220q^{15} + 16777216q^{16} + 128286*a + 47791842q^{17} + -60928*a + 90904512q^{18} + -114129*a + 349993280q^{19} + 258048*a - 57286656q^{20} + 117649*aq^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -952x + -173940$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 5q^{3} + 4q^{4} + q^{6} + 7q^{7} + 3q^{8} + 9q^{9} + 9q^{12} + 3q^{13} + 8q^{14} + 5q^{16} + 3q^{17} + 4q^{18} + 5q^{19} + 2q^{21} + \cdots \in S_{14}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 10q^{2} + 9q^{3} + q^{4} + 2q^{6} + q^{7} + 10q^{8} + q^{9} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{13}$. The form
$f_0 = q + 64q^{2} + -71455084965079460354 + O(11^20)q^{3} + 4096q^{4} + -13972696888007613799*11 + O(11^20)q^{5} + 136124526762834601751 + O(11^20)q^{6} + -117649q^{7} + 262144q^{8} + -317574564268362000514 + O(11^20)q^{9} + 23133755893730911229*11 + O(11^20)q^{10} + -5231466589223644857*11^2 + O(11^21)q^{11} + -33780221301865607549 + O(11^20)q^{12} + 92043363160735822289 + O(11^20)q^{13} + -7529536q^{14} + -30042712725034765960*11 + O(11^20)q^{15} + 16777216q^{16} + 28266995949799157411 + O(11^20)q^{17} + -142272265198367756866 + O(11^20)q^{18} + 39644719207806030323 + O(11^20)q^{19} + 12742206436829207672*11 + O(11^20)q^{20} + -64645620636443787950 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 9q^{2} + q^{3} + 4q^{4} + 4q^{5} + 9q^{6} + 7q^{7} + 3q^{8} + 7q^{9} + 3q^{10} + 4q^{12} + 10q^{13} + 8q^{14} + 4q^{15} + 5q^{16} + 8q^{17} + 8q^{18} + 4q^{19} + 5q^{20} + 7q^{21} + \cdots \in S_{14}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 5q^{3} + 3q^{4} + 5q^{5} + 3q^{6} + 8q^{7} + 4q^{8} + 10q^{9} + 3q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(6)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 64q^{2} + 71455084965079461460 + O(11^20)q^{3} + 4096q^{4} + 153699665768083827319 + O(11^20)q^{5} + -136124526762834530967 + O(11^20)q^{6} + -117649q^{7} + 262144q^{8} + 317574564268363372736 + O(11^20)q^{9} + -254471314831035189599 + O(11^20)q^{10} + 57546132481460396687*11 + O(11^21)q^{11} + 33780221301870137725 + O(11^20)q^{12} + -92043363160727824187 + O(11^20)q^{13} + -7529536q^{14} + 330469839975454997800 + O(11^20)q^{15} + 16777216q^{16} + -28266995949760132579 + O(11^20)q^{17} + 142272265198455579074 + O(11^20)q^{18} + -39644719207681937389 + O(11^20)q^{19} + -140164270804811913512 + O(11^20)q^{20} + 64645620636313668156 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 5q^{3} + 5q^{4} + 6q^{5} + 9q^{6} + 5q^{7} + 9q^{8} + 2q^{9} + 2q^{10} + 3q^{12} + 2q^{13} + 9q^{14} + 8q^{15} + 3q^{16} + 9q^{17} + 8q^{18} + 5q^{19} + 8q^{20} + 3q^{21} + \cdots \in S_{16}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 9q^{3} + 4q^{4} + 10q^{5} + 7q^{6} + 7q^{7} + 8q^{8} + 10q^{9} + 9q^{10} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{4}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -128q^{2} + 7807203393452746695 + O(11^20)q^{3} + 16384q^{4} + -1989857992595405055502 + O(11^21)q^{5} + -326572039429391567759 + O(11^20)q^{6} + -823543q^{7} + -2097152q^{8} + -287284188935161181619 + O(11^20)q^{9} + 3093324947434403663082 + O(11^21)q^{10} + 206047481755422047368*11 + O(11^21)q^{11} + 90721361143400102690 + O(11^20)q^{12} + 2486283182446847126 + O(11^20)q^{13} + 105413504q^{14} + 211963906453239825904 + O(11^20)q^{15} + 268435456q^{16} + 21209118725945449649 + O(11^20)q^{17} + -228873537590169258823 + O(11^20)q^{18} + 102562633712366418640 + O(11^20)q^{19} + 3667903718336976590898 + O(11^21)q^{20} + -96002683779363506428 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 2q^{3} + 5q^{4} + 6q^{5} + 8q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 2q^{10} + 10q^{12} + 6q^{13} + 9q^{14} + q^{15} + 3q^{16} + q^{18} + 2q^{19} + 8q^{20} + 10q^{21} + \cdots \in S_{16}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 3q^{2} + 6q^{3} + 9q^{4} + 8q^{5} + 7q^{6} + 9q^{7} + 5q^{8} + 5q^{9} + 2q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(10)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -128q^{2} + -7807203393452751385 + O(11^20)q^{3} + 16384q^{4} + 1989857992595404977480 + O(11^21)q^{5} + 326572039429392168079 + O(11^20)q^{6} + -823543q^{7} + -2097152q^{8} + 287284188935143819873 + O(11^20)q^{9} + -3093324947434393676266 + O(11^21)q^{10} + -206047481755431118388*11 + O(11^21)q^{11} + -90721361143476943650 + O(11^20)q^{12} + -2486283182720724128 + O(11^20)q^{13} + 105413504q^{14} + -211963906452822617840 + O(11^20)q^{15} + 268435456q^{16} + -1928101702300032707*11 + O(11^20)q^{17} + 228873537592391562311 + O(11^20)q^{18} + -102562633712056193718 + O(11^20)q^{19} + -3667903718338254903346 + O(11^21)q^{20} + 96002683783225923098 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 5q^{4} + 8q^{5} + 5q^{7} + 2q^{8} + 10q^{9} + q^{10} + 4q^{13} + 2q^{14} + 3q^{16} + 3q^{17} + 4q^{18} + 7q^{20} + \cdots \in S_{16}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 9q^{4} + 7q^{5} + 9q^{7} + 6q^{8} + 2q^{9} + q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 128q^{2} + -14819577472334149740*11 + O(11^20)q^{3} + 16384q^{4} + 282160195088452661692 + O(11^20)q^{5} + -974112557920231699*11 + O(11^20)q^{6} + -823543q^{7} + 2097152q^{8} + 37216725750716179755 + O(11^20)q^{9} + -211994755036299800278 + O(11^20)q^{10} + -30396379129595511327*11 + O(11^20)q^{11} + -2368226516960564890*11 + O(11^20)q^{12} + -65621189036914420312 + O(11^20)q^{13} + -105413504q^{14} + 27373917109294621881*11 + O(11^20)q^{15} + 268435456q^{16} + -325802019173778417817 + O(11^20)q^{17} + 54490931563750944233 + O(11^20)q^{18} + 5287040495333281188*11 + O(11^20)q^{19} + -225328847343974067544 + O(11^20)q^{20} + 18154955566308774606*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 8q^{3} + 5q^{4} + 6q^{5} + q^{6} + 6q^{7} + 2q^{8} + 8q^{9} + 9q^{10} + 7q^{12} + 10q^{13} + 9q^{14} + 4q^{15} + 3q^{16} + 5q^{17} + q^{18} + 8q^{19} + 8q^{20} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 2q^{3} + 9q^{4} + 8q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 6q^{9} + 9q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 128q^{2} + aq^{3} + 16384q^{4} + -(301527548504564531876 + O(11^20))*a - 50786220545473251220 + O(11^20)q^{5} + 128*aq^{6} + 823543q^{7} + 2097152q^{8} + -(119543894017838527485 + O(11^20))*a + 242071684065802915050 + O(11^20)q^{9} + -(248776497428339555671 + O(11^20))*a + 226863719505023935850 + O(11^20)q^{10} + (3316803033550209851311*11 + O(11^22))*a + 558648426000206911838*11 + O(11^22)q^{11} + 16384*aq^{12} + -(333495731664369472685 + O(11^20))*a + 104725640414720122469 + O(11^20)q^{13} + 105413504q^{14} + -(93379820821668550600 + O(11^20))*a - 76988548011903891365 + O(11^20)q^{15} + 268435456q^{16} + (31122373792035376489 + O(11^20))*a + 31197315424436046300 + O(11^20)q^{17} + (171631449165548693543 + O(11^20))*a + 38675793525012703154 + O(11^20)q^{18} + -(2950194440308259991241 + O(11^21))*a - 241792668037098997593*11 + O(11^21)q^{19} + -(224141908997142693441 + O(11^20))*a + 110306314542983393157 + O(11^20)q^{20} + 823543*aq^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 119543894017838527485 + O(11^20)x + -242071684065817263957 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 2q^{3} + 5q^{4} + 7q^{5} + 3q^{6} + 6q^{7} + 2q^{8} + 3q^{9} + 5q^{10} + 10q^{12} + 3q^{13} + 9q^{14} + 3q^{15} + 3q^{16} + 4q^{17} + 10q^{18} + 2q^{19} + 2q^{20} + q^{21} + \cdots \in S_{16}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 9q^{2} + 8q^{3} + 4q^{4} + 8q^{5} + 6q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 6q^{10} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{4}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 128q^{2} + 119543894017838530297 + O(11^20)q^{3} + 16384q^{4} + 144166041367141754726 + O(11^20)q^{5} + -171631449165548333607 + O(11^20)q^{6} + 823543q^{7} + 2097152q^{8} + 213110822748698693085 + O(11^20)q^{9} + 289003431815024356501 + O(11^20)q^{10} + -278197395039631288811*11 + O(11^21)q^{11} + 231924339584293601937 + O(11^20)q^{12} + -223966876872820512450 + O(11^20)q^{13} + 105413504q^{14} + 199850018668791469630 + O(11^20)q^{15} + 268435456q^{16} + 48867907097327702751 + O(11^20)q^{17} + -304564480401527662361 + O(11^20)q^{18} + 312680128874906060644 + O(11^20)q^{19} + -8810448967682873927 + O(11^20)q^{20} + -24397303102430082868 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 8q^{3} + 9q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + 5q^{8} + 6q^{10} + 6q^{12} + 2q^{13} + 6q^{14} + 5q^{15} + 4q^{16} + 8q^{17} + 9q^{19} + 7q^{20} + 5q^{21} + \cdots \in S_{18}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 10q^{3} + 5q^{4} + 7q^{5} + 4q^{6} + 5q^{7} + 2q^{8} + 5q^{10} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{6}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 256q^{2} + 31812061040660681393 + O(11^20)q^{3} + 65536q^{4} + 95436183121981591573 + O(11^20)q^{5} + 70887687218414326196 + O(11^20)q^{6} + -5764801q^{7} + 16777216q^{8} + -12781841416893272764*11 + O(11^20)q^{9} + 212663061655127111452 + O(11^20)q^{10} + 24603590073936218861*11 + O(11^21)q^{11} + -17001935265052742251 + O(11^20)q^{12} + 295100372327335945152 + O(11^20)q^{13} + -1475789056q^{14} + 116641405556423788122 + O(11^20)q^{15} + 4294967296q^{16} + -198563495704474277864 + O(11^20)q^{17} + 30439481489707672130*11 + O(11^20)q^{18} + 16194745014267417222 + O(11^20)q^{19} + -51005805824820213569 + O(11^20)q^{20} + 101819364256633126405 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 2q^{3} + 9q^{4} + 6q^{5} + 6q^{6} + 2q^{7} + 5q^{8} + 6q^{9} + 7q^{10} + 7q^{12} + 5q^{13} + 6q^{14} + q^{15} + 4q^{16} + 3q^{17} + 7q^{18} + 3q^{19} + 10q^{20} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 8q^{3} + 5q^{4} + 10q^{5} + 10q^{6} + 6q^{7} + 9q^{8} + 8q^{9} + 7q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 256q^{2} + -31812061040660675751 + O(11^20)q^{3} + 65536q^{4} + -95436183121982479859 + O(11^20)q^{5} + -70887687218412881844 + O(11^20)q^{6} + -5764801q^{7} + 16777216q^{8} + 140600255586121906482 + O(11^20)q^{9} + -212663061655354512668 + O(11^20)q^{10} + -24603590073914263377*11 + O(11^21)q^{11} + 17001935265422496363 + O(11^20)q^{12} + -295100372321456095554 + O(11^20)q^{13} + -1475789056q^{14} + -116641405557314831962 + O(11^20)q^{15} + 4294967296q^{16} + 198563495694349299000 + O(11^20)q^{17} + -334834296311032437462 + O(11^20)q^{18} + -16194744799470256280 + O(11^20)q^{19} + 51005805766605502273 + O(11^20)q^{20} + -101819364289158133647 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 8q^{3} + 9q^{4} + 2q^{5} + 9q^{6} + 9q^{7} + 6q^{8} + 5q^{10} + 6q^{12} + 9q^{13} + 6q^{14} + 5q^{15} + 4q^{16} + 3q^{17} + 2q^{19} + 7q^{20} + 6q^{21} + \cdots \in S_{18}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 10q^{3} + 5q^{4} + 7q^{5} + 4q^{6} + 5q^{7} + 2q^{8} + 5q^{10} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{1}&0\\0&u(5)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -256q^{2} + 253815811816238495084 + O(11^20)q^{3} + 65536q^{4} + -229649504323810436292 + O(11^20)q^{5} + 279901683501266150993 + O(11^20)q^{6} + 5764801q^{7} + -16777216q^{8} + -19467233249146164759*11 + O(11^20)q^{9} + 261023547762750890265 + O(11^20)q^{10} + 23472007902006542545*11 + O(11^20)q^{11} + 329418481459786330299 + O(11^20)q^{12} + 170733113515581954214 + O(11^20)q^{13} + -1475789056q^{14} + -82110775435645210161 + O(11^20)q^{15} + 4294967296q^{16} + -13371441927349415247 + O(11^20)q^{17} + 29725385459839928733*11 + O(11^20)q^{18} + 68847267374141406020 + O(11^20)q^{19} + -219778728940786996941 + O(11^20)q^{20} + 44295492100687023334 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 2q^{3} + 9q^{4} + 6q^{5} + 5q^{6} + 9q^{7} + 6q^{8} + 6q^{9} + 4q^{10} + 7q^{12} + 6q^{13} + 6q^{14} + q^{15} + 4q^{16} + 8q^{17} + 4q^{18} + 8q^{19} + 10q^{20} + 7q^{21} + \cdots \in S_{18}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 8q^{3} + 5q^{4} + 10q^{5} + 10q^{6} + 6q^{7} + 9q^{8} + 8q^{9} + 7q^{10} + q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{6}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -256q^{2} + 41930144096046101557 + O(11^20)q^{3} + 65536q^{4} + 331880770779098262739 + O(11^20)q^{5} + 29883030333158148624 + O(11^20)q^{6} + 5764801q^{7} + -16777216q^{8} + 31906908150322110923 + O(11^20)q^{9} + -194977957946594101858 + O(11^20)q^{10} + 6936549970480912135*11 + O(11^20)q^{11} + -249805821030325946533 + O(11^20)q^{12} + 298835465118840732055 + O(11^20)q^{13} + -1475789056q^{14} + -218455948637300923350 + O(11^20)q^{15} + 4294967296q^{16} + 80359725564331409721 + O(11^20)q^{17} + -95168547291740285876 + O(11^20)q^{18} + 94848546895699174047 + O(11^20)q^{19} + 130857609318649394774 + O(11^20)q^{20} + -136564238149004024143 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 3q^{4} + 2q^{5} + 8q^{7} + 4q^{8} + 7q^{9} + 10q^{10} + 6q^{13} + 7q^{14} + 9q^{16} + 2q^{17} + 2q^{18} + 6q^{20} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 9q^{4} + 7q^{5} + 9q^{7} + 6q^{8} + 2q^{9} + q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{8}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -512q^{2} + -28733405673548700037*11 + O(11^20)q^{3} + 262144q^{4} + -317102298187116756587 + O(11^20)q^{5} + -27837093210971237187*11 + O(11^20)q^{6} + 40353607q^{7} + -134217728q^{8} + 69049025165309437981 + O(11^20)q^{9} + 223627893056817155103 + O(11^20)q^{10} + 20969592249212352982*11 + O(11^20)q^{11} + 2523649536684153941*11 + O(11^20)q^{12} + -164713839026173572008 + O(11^20)q^{13} + -20661046784q^{14} + -18116445064347617715*11 + O(11^20)q^{15} + 68719476736q^{16} + 217349377416645842535 + O(11^20)q^{17} + 302648846787248241381 + O(11^20)q^{18} + 122717808346203518*11^2 + O(11^20)q^{19} + -129982106555181848566 + O(11^20)q^{20} + -18033049919959533214*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 7q^{3} + 3q^{4} + 10q^{5} + 2q^{6} + 3q^{7} + 4q^{8} + q^{9} + 6q^{10} + 10q^{12} + 10q^{13} + 4q^{14} + 4q^{15} + 9q^{16} + 10q^{17} + 5q^{18} + 4q^{19} + 8q^{20} + 10q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 9q^{2} + 8q^{3} + 4q^{4} + 8q^{5} + 6q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 6q^{10} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{1}&0\\0&u(5)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -512q^{2} + -88543766322218070626 + O(11^20)q^{3} + 262144q^{4} + -289792308788191874776 + O(11^20)q^{5} + 260158696494131544045 + O(11^20)q^{6} + -40353607q^{7} + -134217728q^{8} + 317791722004941507137 + O(11^20)q^{9} + -304086780541522148109 + O(11^20)q^{10} + 220225206379366684713*11 + O(11^21)q^{11} + 3246391651531270758 + O(11^20)q^{12} + -38112882514750438759 + O(11^20)q^{13} + 20661046784q^{14} + 226042674673868636973 + O(11^20)q^{15} + 68719476736q^{16} + -71712117644626233798 + O(11^20)q^{17} + 96137107149470572498 + O(11^20)q^{18} + -52026293102723072624 + O(11^20)q^{19} + 287182807837977706377 + O(11^20)q^{20} + -313869559253647831545 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 6q^{3} + 3q^{4} + 7q^{5} + 8q^{6} + 3q^{7} + 4q^{8} + 10q^{9} + 2q^{10} + 7q^{12} + 4q^{13} + 4q^{14} + 9q^{15} + 9q^{16} + 7q^{17} + 6q^{18} + 5q^{19} + 10q^{20} + 7q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + 2q^{3} + 9q^{4} + 8q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 6q^{9} + 9q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{8}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -512q^{2} + aq^{3} + 262144q^{4} + -(322513104493254508046 + O(11^20))*a + 244564215959920200276 + O(11^20)q^{5} + -512*aq^{6} + -40353607q^{7} + -134217728q^{8} + (88543766322218101918 + O(11^20))*a + 27901650363128691780 + O(11^20)q^{9} + (302960742069105865307 + O(11^20))*a - 85379514022980829926 + O(11^20)q^{10} + -(182922628804035616394*11 + O(11^21))*a - 255239115454051740229*11 + O(11^21)q^{11} + 262144*aq^{12} + (131754868830797288158 + O(11^20))*a - 260625834194036769220 + O(11^20)q^{13} + 20661046784q^{14} + (45228092828269722046 + O(11^20))*a - 218578168036968076556 + O(11^20)q^{15} + 68719476736q^{16} + -(39640398502677007787 + O(11^20))*a + 24148215320092318013 + O(11^20)q^{17} + -(260158696494147565549 + O(11^20))*a - 157895092338129998139 + O(11^20)q^{18} + (2878242378016569516996 + O(11^21))*a + 212462003907491390601*11 + O(11^21)q^{19} + (289348890039159088247 + O(11^20))*a - 14438490850215675953 + O(11^20)q^{20} + -40353607*aq^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -88543766322218101918 + O(11^20)x + -27901650364290953247 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 7q^{3} + 3q^{4} + 7q^{5} + 9q^{6} + 8q^{7} + 7q^{8} + q^{9} + 9q^{10} + 10q^{12} + 9q^{13} + 4q^{14} + 5q^{15} + 9q^{16} + 6q^{18} + 4q^{19} + 10q^{20} + q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 3q^{2} + 6q^{3} + 9q^{4} + 8q^{5} + 7q^{6} + 9q^{7} + 5q^{8} + 5q^{9} + 2q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{8}&0\\0&u(10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 512q^{2} + -212435844131104615608 + O(11^20)q^{3} + 262144q^{4} + -166350572082550052839 + O(11^20)q^{5} + 218346983949158299266 + O(11^20)q^{6} + 40353607q^{7} + 134217728q^{8} + -151640659715407733573 + O(11^20)q^{9} + 267756450169494114959 + O(11^20)q^{10} + -29016309934628874733*11 + O(11^20)q^{11} + 117156623164087696826 + O(11^20)q^{12} + -30163309034502243360 + O(11^20)q^{13} + 20661046784q^{14} + 49241021454514169087 + O(11^20)q^{15} + 68719476736q^{16} + 20458140033638085737*11 + O(11^20)q^{17} + -273768357044358531261 + O(11^20)q^{18} + -183025940344440506826 + O(11^20)q^{19} + -149696479461255017996 + O(11^20)q^{20} + -100601974791207067903 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + q^{3} + 3q^{4} + 7q^{5} + 6q^{6} + 8q^{7} + 7q^{8} + 8q^{9} + 9q^{10} + 3q^{12} + 3q^{13} + 4q^{14} + 7q^{15} + 9q^{16} + 6q^{17} + 4q^{18} + 10q^{19} + 10q^{20} + 8q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 2q^{2} + 9q^{3} + 4q^{4} + 10q^{5} + 7q^{6} + 7q^{7} + 8q^{8} + 10q^{9} + 9q^{10} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(4)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 512q^{2} + 98397500895388515283 + O(11^20)q^{3} + 262144q^{4} + 80466606255616733555 + O(11^20)q^{5} + -76729161503080865179 + O(11^20)q^{6} + 40353607q^{7} + 134217728q^{8} + 164242079669970465545 + O(11^20)q^{9} + 161152711989607018899 + O(11^20)q^{10} + 24560838001429294949*11 + O(11^20)q^{11} + -265830983488922437990 + O(11^20)q^{12} + -83754095754752859587 + O(11^20)q^{13} + 20661046784q^{14} + 287074315542583532861 + O(11^20)q^{15} + 68719476736q^{16} + 313109924524578582550 + O(11^20)q^{17} + -1804575545122791085 + O(11^20)q^{18} + -94370087850337774586 + O(11^20)q^{19} + -238060838026087455435 + O(11^20)q^{20} + -175176380507302319203 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + z^112q^{3} + q^{4} + z^97q^{5} + z^112q^{6} + 10q^{7} + q^{8} + z^2q^{9} + z^97q^{10} + z^112q^{12} + z^43q^{13} + 10q^{14} + z^89q^{15} + q^{16} + z^90q^{17} + z^2q^{18} + z^91q^{19} + z^97q^{20} + z^52q^{21} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{2} + z^16q^{3} + 3q^{4} + z^49q^{5} + z^4q^{6} + 3q^{7} + 7q^{8} + z^50q^{9} + z^37q^{10} + z^91q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{0}&0\\0&u(z^91)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1024q^{2} + aq^{3} + 1048576q^{4} + -(178039985429450202359 + O(11^20))*a - 248156152481185400376 + O(11^20)q^{5} + 1024*aq^{6} + -282475249q^{7} + 1073741824q^{8} + (306002974327305080828 + O(11^20))*a + 3115271135234090870 + O(11^20)q^{9} + (2303546966755277855 + O(11^20))*a + 187597943773833492954 + O(11^20)q^{10} + -(1771554646678636890*11 + O(11^20))*a + 21613591510010758474*11 + O(11^20)q^{11} + 1048576*aq^{12} + (270889201092463293959 + O(11^20))*a + 183042683592197536407 + O(11^20)q^{13} + -289254654976q^{14} + -(181670430045426612893 + O(11^20))*a + 122100611589142006749 + O(11^20)q^{15} + 1099511627776q^{16} + -(239133139984143310563 + O(11^20))*a + 122187333694336029953 + O(11^20)q^{17} + -(154451927412561519794 + O(11^20))*a - 173712332183090995125 + O(11^20)q^{18} + -(36171358559449276390 + O(11^20))*a + 306516068180385297832 + O(11^20)q^{19} + -(332167885772835513284 + O(11^20))*a - 306204126306665846590 + O(11^20)q^{20} + -282475249*aq^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -306002974327305080828 + O(11^20)x + -3115271145694444073 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + z^32q^{3} + q^{4} + z^107q^{5} + z^32q^{6} + 10q^{7} + q^{8} + z^22q^{9} + z^107q^{10} + z^32q^{12} + z^113q^{13} + 10q^{14} + z^19q^{15} + q^{16} + z^30q^{17} + z^22q^{18} + z^41q^{19} + z^107q^{20} + z^92q^{21} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 6q^{2} + z^56q^{3} + 3q^{4} + z^59q^{5} + z^44q^{6} + 3q^{7} + 7q^{8} + z^70q^{9} + z^47q^{10} + z^41q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^79)\omega^{0}&0\\0&u(z^41)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1024q^{2} + aq^{3} + 1048576q^{4} + -(178039985429450202359 + O(11^20))*a - 248156152481185400376 + O(11^20)q^{5} + 1024*aq^{6} + -282475249q^{7} + 1073741824q^{8} + (306002974327305080828 + O(11^20))*a + 3115271135234090870 + O(11^20)q^{9} + (2303546966755277855 + O(11^20))*a + 187597943773833492954 + O(11^20)q^{10} + -(1771554646678636890*11 + O(11^20))*a + 21613591510010758474*11 + O(11^20)q^{11} + 1048576*aq^{12} + (270889201092463293959 + O(11^20))*a + 183042683592197536407 + O(11^20)q^{13} + -289254654976q^{14} + -(181670430045426612893 + O(11^20))*a + 122100611589142006749 + O(11^20)q^{15} + 1099511627776q^{16} + -(239133139984143310563 + O(11^20))*a + 122187333694336029953 + O(11^20)q^{17} + -(154451927412561519794 + O(11^20))*a - 173712332183090995125 + O(11^20)q^{18} + -(36171358559449276390 + O(11^20))*a + 306516068180385297832 + O(11^20)q^{19} + -(332167885772835513284 + O(11^20))*a - 306204126306665846590 + O(11^20)q^{20} + -282475249*aq^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -306002974327305080828 + O(11^20)x + -3115271145694444073 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 4q^{3} + q^{4} + 3q^{5} + 7q^{6} + q^{7} + 10q^{8} + 2q^{9} + 8q^{10} + 4q^{12} + 5q^{13} + 10q^{14} + q^{15} + q^{16} + 5q^{17} + 9q^{18} + 6q^{19} + 3q^{20} + 4q^{21} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 5q^{3} + 3q^{4} + 5q^{5} + 3q^{6} + 8q^{7} + 4q^{8} + 10q^{9} + 3q^{10} + 6q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1024q^{2} + -243127244563705540012 + O(11^20)q^{3} + 1048576q^{4} + -59655181185898158231 + O(11^20)q^{5} + 44800308187269567918 + O(11^20)q^{6} + 282475249q^{7} + -1073741824q^{8} + 231597047784313100639 + O(11^20)q^{9} + -133344004503246808747 + O(11^20)q^{10} + -339997677635748457*11 + O(11^19)q^{11} + -128515928349956922364 + O(11^20)q^{12} + 134558513680610573955 + O(11^20)q^{13} + -289254654976q^{14} + -144129109102847490434 + O(11^20)q^{15} + 1099511627776q^{16} + 144359684245610315575 + O(11^20)q^{17} + 325371280057068193617 + O(11^20)q^{18} + 31196334320133697908 + O(11^20)q^{19} + -23988359984949710875 + O(11^20)q^{20} + -13624314745968555966 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 9q^{3} + q^{4} + 2q^{6} + q^{7} + 10q^{8} + q^{9} + 9q^{12} + 7q^{13} + 10q^{14} + q^{16} + 6q^{17} + 10q^{18} + 2q^{19} + 9q^{21} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 10q^{2} + 9q^{3} + q^{4} + 2q^{6} + q^{7} + 10q^{8} + q^{9} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{1}$. The form
$f_0 = q + -1024q^{2} + aq^{3} + 1048576q^{4} + (82522415730337645153 + O(11^20))*a + 58771318789010497346 + O(11^20)q^{5} + -1024*aq^{6} + 282475249q^{7} + -1073741824q^{8} + (243127244563705387376 + O(11^20))*a + 176033703029438919702 + O(11^20)q^{9} + (263545653636812522654 + O(11^20))*a - 307080890948908463415 + O(11^20)q^{10} + -(236775406310654129615*11^-1 + O(11^19))*a - 244195287253577509858*11^-1 + O(11^19)q^{11} + 1048576*aq^{12} + -(20234923239646134248 + O(11^20))*a - 13555505478385604938*11 + O(11^20)q^{13} + -289254654976q^{14} + (883862396903721559 + O(11^20))*a + 43495215476657951280 + O(11^20)q^{15} + 1099511627776q^{16} + (156656221204214568343 + O(11^20))*a + 280995984318022920240 + O(11^20)q^{17} + -(44800308187113268654 + O(11^20))*a + 38486739780628691020 + O(11^20)q^{18} + (323304623022660072 + O(11^20))*a - 240349055759379841301 + O(11^20)q^{19} + -(98001356139459508095 + O(11^20))*a + 276584698176742240093 + O(11^20)q^{20} + 282475249*aq^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -243127244563705387376 + O(11^20)x + -176033703039899272905 + O(11^20)$.
The slope of $f_0$ is $1/2$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 9q^{2} + 8q^{3} + 4q^{4} + 8q^{5} + 6q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 6q^{10} + 5q^{11} + 10q^{12} + 7q^{13} + 3q^{14} + 9q^{15} + 5q^{16} + 8q^{17} + 3q^{18} + 3q^{19} + 10q^{20} + 10q^{21} + \cdots \in S_{14}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{2} + 9q^{3} + 4q^{4} + 10q^{5} + 7q^{6} + 7q^{7} + 8q^{8} + 10q^{9} + 9q^{10} + 4q^{11} + 3q^{12} + q^{13} + 3q^{14} + 2q^{15} + 5q^{16} + 7q^{17} + 9q^{18} + 2q^{19} + 7q^{20} + 8q^{21} + \cdots \in S_{14}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 8q^{3} + 5q^{4} + 10q^{5} + 10q^{6} + 6q^{7} + 9q^{8} + 8q^{9} + 7q^{10} + q^{11} + 7q^{12} + 3q^{13} + 2q^{14} + 3q^{15} + 3q^{16} + 5q^{17} + 10q^{18} + q^{19} + 6q^{20} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 10q^{3} + 5q^{4} + 7q^{5} + 4q^{6} + 5q^{7} + 2q^{8} + 5q^{10} + 5q^{11} + 6q^{12} + q^{13} + 2q^{14} + 4q^{15} + 3q^{16} + 5q^{17} + 8q^{19} + 2q^{20} + 6q^{21} + \cdots \in S_{16}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 6q^{3} + 9q^{4} + 8q^{5} + 7q^{6} + 9q^{7} + 5q^{8} + 5q^{9} + 2q^{10} + 10q^{11} + 10q^{12} + 10q^{13} + 5q^{14} + 4q^{15} + 4q^{16} + 4q^{18} + 6q^{19} + 6q^{20} + 10q^{21} + \cdots \in S_{18}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + 2q^{3} + 9q^{4} + 8q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 6q^{9} + 9q^{10} + 6q^{11} + 7q^{12} + 2q^{13} + 5q^{14} + 5q^{15} + 4q^{16} + 3q^{17} + 4q^{18} + 2q^{19} + 6q^{20} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + 9q^{4} + 7q^{5} + 9q^{7} + 6q^{8} + 2q^{9} + q^{10} + 7q^{11} + 3q^{13} + 6q^{14} + 4q^{16} + 4q^{17} + 5q^{18} + 8q^{20} + \cdots \in S_{18}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 5q^{3} + 3q^{4} + 5q^{5} + 3q^{6} + 8q^{7} + 4q^{8} + 10q^{9} + 3q^{10} + 6q^{11} + 4q^{12} + 8q^{13} + 7q^{14} + 3q^{15} + 9q^{16} + 10q^{17} + 6q^{18} + 9q^{19} + 4q^{20} + 7q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + z^16q^{3} + 3q^{4} + z^49q^{5} + z^4q^{6} + 3q^{7} + 7q^{8} + z^50q^{9} + z^37q^{10} + z^91q^{11} + z^112q^{12} + z^31q^{13} + 7q^{14} + z^65q^{15} + 9q^{16} + z^102q^{17} + z^38q^{18} + z^55q^{19} + z^25q^{20} + z^112q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + z^56q^{3} + 3q^{4} + z^59q^{5} + z^44q^{6} + 3q^{7} + 7q^{8} + z^70q^{9} + z^47q^{10} + z^41q^{11} + z^32q^{12} + z^101q^{13} + 7q^{14} + z^115q^{15} + 9q^{16} + z^42q^{17} + z^58q^{18} + z^5q^{19} + z^35q^{20} + z^32q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 5q^{3} + 3q^{4} + 10q^{5} + 8q^{6} + 8q^{7} + 7q^{8} + 10q^{9} + 5q^{10} + q^{11} + 4q^{12} + 7q^{13} + 4q^{14} + 6q^{15} + 9q^{16} + 3q^{17} + 5q^{18} + 8q^{19} + 8q^{20} + 7q^{21} + \cdots \in S_{20}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + q^{4} + 7q^{5} + 10q^{7} + 10q^{8} + 8q^{9} + 4q^{10} + 10q^{11} + 2q^{13} + q^{14} + q^{16} + 7q^{17} + 3q^{18} + 5q^{19} + 7q^{20} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + 2q^{3} + q^{4} + 2q^{5} + 9q^{6} + 10q^{7} + 10q^{8} + q^{9} + 9q^{10} + q^{11} + 2q^{12} + 7q^{13} + q^{14} + 4q^{15} + q^{16} + 10q^{18} + 4q^{19} + 2q^{20} + 9q^{21} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 6q^{3} + q^{4} + 5q^{5} + 6q^{6} + q^{7} + q^{8} + 5q^{10} + q^{11} + 6q^{12} + 3q^{13} + q^{14} + 8q^{15} + q^{16} + q^{17} + 10q^{19} + 5q^{20} + 6q^{21} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 3q^{3} + q^{4} + 8q^{5} + 3q^{6} + q^{7} + q^{8} + 6q^{9} + 8q^{10} + q^{11} + 3q^{12} + 6q^{13} + q^{14} + 2q^{15} + q^{16} + 6q^{17} + 6q^{18} + 2q^{19} + 8q^{20} + 3q^{21} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + q^{4} + 2q^{5} + 10q^{7} + q^{8} + 8q^{9} + 2q^{10} + 10q^{11} + 2q^{13} + 10q^{14} + q^{16} + 2q^{17} + 8q^{18} + 2q^{20} + \cdots \in S_{22}(\Gamma_0(14);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 15, \quad \ell = 11}$ \quad (55 forms)}

\begin{enumerate}
\item  Consider
$f = q + 8q^{3} + 3q^{4} + 6q^{5} + 2q^{7} + 9q^{9} + 2q^{12} + 4q^{15} + 9q^{16} + 6q^{17} + q^{19} + 7q^{20} + 5q^{21} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{3} + 5q^{4} + 2q^{5} + 7q^{7} + 5q^{9} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -9751629919976711636*11 + O(11^20)q^{2} + -729q^{3} + -156251017811293567302 + O(11^20)q^{4} + -15625q^{5} + 14483719646935412888*11 + O(11^20)q^{6} + -91184784065611439971 + O(11^20)q^{7} + -683186076967310296*11 + O(11^20)q^{8} + 531441q^{9} + 21923192635484501619*11 + O(11^20)q^{10} + -16230108369330856444*11 + O(11^20)q^{11} + 212242840830369008189 + O(11^20)q^{12} + 22275811034683964292*11 + O(11^20)q^{13} + 22127209722227597018*11 + O(11^20)q^{14} + 11390625q^{15} + -296819141045490644759 + O(11^20)q^{16} + 189957442124460364030 + O(11^20)q^{17} + 21891024959800512991*11 + O(11^20)q^{18} + -139074851511684630068 + O(11^20)q^{19} + 12421691201715703321 + O(11^20)q^{20} + -128541914492701172040 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^37q^{2} + 3q^{3} + z^93q^{4} + 6q^{5} + z^13q^{6} + z^71q^{7} + 7q^{8} + 9q^{9} + z^25q^{10} + z^69q^{12} + z^49q^{13} + 6q^{14} + 7q^{15} + z^87q^{16} + 3q^{17} + z^109q^{18} + 3q^{19} + z^81q^{20} + z^47q^{21} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^13q^{2} + 4q^{3} + z^45q^{4} + 2q^{5} + z^37q^{6} + z^23q^{7} + 2q^{8} + 5q^{9} + z^25q^{10} + z^55q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^65)\omega^{1}&0\\0&u(z^55)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 729q^{3} + 131*a - 5460q^{4} + -15625q^{5} + 729*aq^{6} + 1624*a + 141764q^{7} + 29*11^2*a + 357892q^{8} + 531441q^{9} + -15625*aq^{10} + 3232*11*a + 72208*11q^{11} + 95499*a - 3980340q^{12} + 3016*a + 62094*11q^{13} + 32228*11*a + 4436768q^{14} + -11390625q^{15} + -255581*a + 54314908q^{16} + -55400*11*a + 63991502q^{17} + 531441*aq^{18} + -378808*11*a + 292168836q^{19} + -2046875*a + 85312500q^{20} + 1183896*a + 103345956q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -131x + -2732$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^47q^{2} + 3q^{3} + z^63q^{4} + 6q^{5} + z^23q^{6} + z^61q^{7} + 7q^{8} + 9q^{9} + z^35q^{10} + z^39q^{12} + z^59q^{13} + 6q^{14} + 7q^{15} + z^117q^{16} + 3q^{17} + z^119q^{18} + 3q^{19} + z^51q^{20} + z^37q^{21} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^23q^{2} + 4q^{3} + z^15q^{4} + 2q^{5} + z^47q^{6} + z^13q^{7} + 2q^{8} + 5q^{9} + z^35q^{10} + z^5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^115)\omega^{1}&0\\0&u(z^5)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 729q^{3} + 131*a - 5460q^{4} + -15625q^{5} + 729*aq^{6} + 1624*a + 141764q^{7} + 29*11^2*a + 357892q^{8} + 531441q^{9} + -15625*aq^{10} + 3232*11*a + 72208*11q^{11} + 95499*a - 3980340q^{12} + 3016*a + 62094*11q^{13} + 32228*11*a + 4436768q^{14} + -11390625q^{15} + -255581*a + 54314908q^{16} + -55400*11*a + 63991502q^{17} + 531441*aq^{18} + -378808*11*a + 292168836q^{19} + -2046875*a + 85312500q^{20} + 1183896*a + 103345956q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -131x + -2732$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 8q^{3} + 7q^{4} + 5q^{5} + 6q^{6} + 2q^{8} + 9q^{9} + q^{10} + q^{12} + 7q^{13} + 7q^{15} + 6q^{16} + q^{17} + 4q^{18} + 10q^{19} + 2q^{20} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 10q^{2} + 10q^{3} + 10q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 10q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{2}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -16435229468117994987 + O(11^20)q^{2} + -729q^{3} + 321336111263610465447 + O(11^20)q^{4} + 15625q^{5} + -128217626528061820095 + O(11^20)q^{6} + 6527467962427583892*11 + O(11^20)q^{7} + -305467132190083741634 + O(11^20)q^{8} + 531441q^{9} + 190037624894251842907 + O(11^20)q^{10} + -176088697983489315312*11 + O(11^21)q^{11} + -137026874641146108915 + O(11^20)q^{12} + 302802986877339314021 + O(11^20)q^{13} + -30539046920140088276*11 + O(11^20)q^{14} + -11390625q^{15} + 293540100060264551599 + O(11^20)q^{16} + -175657480920550390926 + O(11^20)q^{17} + -41599556668774429684 + O(11^20)q^{18} + -98819649541076636833 + O(11^20)q^{19} + 143526312218173942312 + O(11^20)q^{20} + 11884910366625953430*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^73q^{2} + 8q^{3} + z^103q^{4} + 5q^{5} + z^109q^{6} + z^27q^{7} + z^101q^{8} + 9q^{9} + zq^{10} + z^19q^{12} + z^85q^{13} + z^100q^{14} + 7q^{15} + z^38q^{16} + z^59q^{17} + z^25q^{18} + z^3q^{19} + z^31q^{20} + z^63q^{21} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^49q^{2} + 7q^{3} + z^55q^{4} + 9q^{5} + z^13q^{6} + z^99q^{7} + z^29q^{8} + 5q^{9} + zq^{10} + z^44q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^76)\omega^{1}&0\\0&u(z^44)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -729q^{3} + (16435229468117994922 + O(11^20))*a + 74203809231499201161 + O(11^20)q^{4} + 15625q^{5} + -729*aq^{6} + (179065013231154110692 + O(11^20))*a - 146869377507410534228 + O(11^20)q^{7} + (277210074437450350126 + O(11^20))*a - 113997635519528443410 + O(11^20)q^{8} + 531441q^{9} + 15625*aq^{10} + (146822651952784174193*11 + O(11^21))*a + 316478084322265966920*11 + O(11^21)q^{11} + (128217626528061867480 + O(11^20))*a - 274577335158116910289 + O(11^20)q^{12} + (602555301763172103865 + O(11^21))*a - 38857563297046352189*11 + O(11^21)q^{13} + (75067229920706614024 + O(11^20))*a - 201912429175003893001 + O(11^20)q^{14} + -11390625q^{15} + (332063951398997197529 + O(11^20))*a - 82162044654808286690 + O(11^20)q^{16} + (190979793642920608565 + O(11^20))*a - 76308082936306932757 + O(11^20)q^{17} + 531441*aq^{18} + -(85611631633033831717 + O(11^20))*a - 220973960899392707388 + O(11^20)q^{19} + -(190037624894252858532 + O(11^20))*a + 286277973374122287302 + O(11^20)q^{20} + -(24895628594704909474 + O(11^20))*a + 100527008625237989253 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -16435229468117994922 + O(11^20)x + -74203809231499209353 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^83q^{2} + 8q^{3} + z^53q^{4} + 5q^{5} + z^119q^{6} + z^57q^{7} + z^31q^{8} + 9q^{9} + z^11q^{10} + z^89q^{12} + z^95q^{13} + z^20q^{14} + 7q^{15} + z^58q^{16} + z^49q^{17} + z^35q^{18} + z^33q^{19} + z^101q^{20} + z^93q^{21} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^59q^{2} + 7q^{3} + z^5q^{4} + 9q^{5} + z^23q^{6} + z^9q^{7} + z^79q^{8} + 5q^{9} + z^11q^{10} + z^4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^116)\omega^{1}&0\\0&u(z^4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -729q^{3} + (16435229468117994922 + O(11^20))*a + 74203809231499201161 + O(11^20)q^{4} + 15625q^{5} + -729*aq^{6} + (179065013231154110692 + O(11^20))*a - 146869377507410534228 + O(11^20)q^{7} + (277210074437450350126 + O(11^20))*a - 113997635519528443410 + O(11^20)q^{8} + 531441q^{9} + 15625*aq^{10} + (146822651952784174193*11 + O(11^21))*a + 316478084322265966920*11 + O(11^21)q^{11} + (128217626528061867480 + O(11^20))*a - 274577335158116910289 + O(11^20)q^{12} + (602555301763172103865 + O(11^21))*a - 38857563297046352189*11 + O(11^21)q^{13} + (75067229920706614024 + O(11^20))*a - 201912429175003893001 + O(11^20)q^{14} + -11390625q^{15} + (332063951398997197529 + O(11^20))*a - 82162044654808286690 + O(11^20)q^{16} + (190979793642920608565 + O(11^20))*a - 76308082936306932757 + O(11^20)q^{17} + 531441*aq^{18} + -(85611631633033831717 + O(11^20))*a - 220973960899392707388 + O(11^20)q^{19} + -(190037624894252858532 + O(11^20))*a + 286277973374122287302 + O(11^20)q^{20} + -(24895628594704909474 + O(11^20))*a + 100527008625237989253 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -16435229468117994922 + O(11^20)x + -74203809231499209353 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{3} + q^{4} + 8q^{5} + 5q^{7} + 4q^{9} + 9q^{12} + 3q^{13} + 6q^{15} + q^{16} + 4q^{17} + 8q^{19} + 8q^{20} + q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{3} + 4q^{4} + 7q^{5} + 9q^{7} + 3q^{9} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(4)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 260488084415886969*11^2 + O(11^20)q^{2} + 2187q^{3} + -270761233335006997175 + O(11^20)q^{4} + -78125q^{5} + 2575874641337190141*11^2 + O(11^20)q^{6} + -231162659124525621202 + O(11^20)q^{7} + 1906158953185967235*11^2 + O(11^20)q^{8} + 4782969q^{9} + -1334227609602232665*11^2 + O(11^20)q^{10} + 29835902138823472635*11 + O(11^20)q^{11} + -134821763007494724845 + O(11^20)q^{12} + -178454116099437923511 + O(11^20)q^{13} + -708594907426405441*11^2 + O(11^20)q^{14} + -170859375q^{15} + -139825144528303425047 + O(11^20)q^{16} + -2261985481387611453129 + O(11^21)q^{17} + 1241602036804348114*11^2 + O(11^20)q^{18} + -315186207215027237290 + O(11^20)q^{19} + -56736367062715010168 + O(11^20)q^{20} + -317489310984966658823 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 2q^{3} + 4q^{4} + 8q^{5} + q^{6} + 8q^{7} + 8q^{8} + 4q^{9} + 4q^{10} + 8q^{12} + 5q^{13} + 4q^{14} + 5q^{15} + 8q^{16} + 5q^{17} + 2q^{18} + 9q^{19} + 10q^{20} + 5q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 8q^{3} + q^{4} + 6q^{5} + 2q^{6} + 9q^{7} + q^{8} + 9q^{9} + 7q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{4}&0\\0&u(9)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -133846224224865215552 + O(11^20)q^{2} + -2187q^{3} + -45817125007865263246 + O(11^20)q^{4} + -78125q^{5} + 75444584116622409789 + O(11^20)q^{6} + -312623324206051918513 + O(11^20)q^{7} + -210129644913620890175 + O(11^20)q^{8} + 4782969q^{9} + 183096330814741988857 + O(11^20)q^{10} + -29007296767092437525*11 + O(11^20)q^{11} + -37696852750110651947 + O(11^20)q^{12} + 290743075577355649704 + O(11^20)q^{13} + 119691740227359379347 + O(11^20)q^{14} + 170859375q^{15} + -81173452332401553215 + O(11^20)q^{16} + -318966492302716197792 + O(11^20)q^{17} + -173556704576007954298 + O(11^20)q^{18} + -177831421681845354469 + O(11^20)q^{19} + -239831796678117864771 + O(11^20)q^{20} + 193215187154576439715 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 2q^{3} + 2q^{4} + 8q^{5} + 2q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{10} + 4q^{12} + 4q^{13} + 4q^{14} + 5q^{15} + 5q^{16} + q^{17} + 4q^{18} + 10q^{19} + 5q^{20} + 8q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 9q^{2} + 6q^{3} + 8q^{4} + 7q^{5} + 10q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 8q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 140340332360087566829 + O(11^20)q^{2} + -2187q^{3} + -286498081121202048646 + O(11^20)q^{4} + -78125q^{5} + -150309182264144459367 + O(11^20)q^{6} + 306181955836661377334 + O(11^20)q^{7} + -106880417729400798029 + O(11^20)q^{8} + 4782969q^{9} + -281798215910688566928 + O(11^20)q^{10} + 16372805778670364971*11 + O(11^20)q^{11} + 241058129855511822671 + O(11^20)q^{12} + 305284089591231202478 + O(11^20)q^{13} + 101758978916901986387 + O(11^20)q^{14} + 170859375q^{15} + 72214326856386649324 + O(11^20)q^{16} + -48988938352072850966 + O(11^20)q^{17} + -248565910337911863660 + O(11^20)q^{18} + 331434853145305979387 + O(11^20)q^{19} + 270256187638544351480 + O(11^20)q^{20} + -233692456881223074463 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^112q^{2} + 9q^{3} + z^21q^{4} + 3q^{5} + z^64q^{6} + z^71q^{7} + z^82q^{8} + 4q^{9} + z^88q^{10} + z^93q^{12} + z^10q^{13} + z^63q^{14} + 5q^{15} + z^64q^{16} + z^10q^{17} + z^16q^{18} + z^68q^{19} + z^117q^{20} + z^23q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^64q^{2} + 5q^{3} + z^45q^{4} + 4q^{5} + z^112q^{6} + z^95q^{7} + z^58q^{8} + 3q^{9} + z^88q^{10} + z^91q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{1}&0\\0&u(z^91)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 2187q^{3} + -(1173233073618776421 + O(11^20))*a + 170130902257686544534 + O(11^20)q^{4} + 78125q^{5} + 2187*aq^{6} + -(14078796883425318632 + O(11^20))*a + 23320842294559336013 + O(11^20)q^{7} + -(224087517061186304777 + O(11^20))*a + 203002574455249602762 + O(11^20)q^{8} + 4782969q^{9} + 78125*aq^{10} + (272190073079556157872*11 + O(11^21))*a + 221880377237764170119*11 + O(11^21)q^{11} + (125139247725976004077 + O(11^20))*a + 45536039854787807705 + O(11^20)q^{12} + -(126998270124211985643 + O(11^20))*a + 230983493093805052022 + O(11^20)q^{13} + -(162591953484007680519 + O(11^20))*a + 38203319611966739580 + O(11^20)q^{14} + 170859375q^{15} + -(1941196452709988715100 + O(11^21))*a + 18877802782488392113*11 + O(11^21)q^{16} + -(246406075522738024543 + O(11^20))*a + 20770961843675563051 + O(11^20)q^{17} + 4782969*aq^{18} + -(183763481661406868834 + O(11^20))*a - 97056046915720777000 + O(11^20)q^{19} + -(164834565638746639289 + O(11^20))*a - 44911000826810065407 + O(11^20)q^{20} + (156170982846588575062 + O(11^20))*a - 126317516673292838845 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 1173233073618776421 + O(11^20)x + -170130902257686577302 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 9q^{3} + 5q^{4} + 3q^{5} + 7q^{6} + 8q^{7} + q^{8} + 4q^{9} + 6q^{10} + q^{12} + 5q^{14} + 5q^{15} + 7q^{16} + 4q^{17} + 8q^{18} + 5q^{19} + 4q^{20} + 6q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 3q^{6} + 9q^{7} + 7q^{8} + 9q^{9} + 5q^{10} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{4}&0\\0&u(8)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1173233073618776612 + O(11^20)q^{2} + 2187q^{3} + 53956614803499779992 + O(11^20)q^{4} + 78125q^{5} + -125139247725975586360 + O(11^20)q^{6} + 139271111189451160554 + O(11^20)q^{7} + -182304461358787128305 + O(11^20)q^{8} + 4782969q^{9} + 164834565638761561164 + O(11^20)q^{10} + -288011801271806825827*11 + O(11^21)q^{11} + 271867462056017232329 + O(11^20)q^{12} + -14371612738478646004*11 + O(11^20)q^{13} + -96327995460906973374 + O(11^20)q^{14} + 170859375q^{15} + -56075959961317515454 + O(11^20)q^{16} + 327371976706545664227 + O(11^20)q^{17} + 129713160843316375487 + O(11^20)q^{18} + 12395119427673751554 + O(11^20)q^{19} + -90936724000705778466 + O(11^20)q^{20} + -169827533119996036455 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^32q^{2} + 9q^{3} + z^111q^{4} + 3q^{5} + z^104q^{6} + z^61q^{7} + z^62q^{8} + 4q^{9} + z^8q^{10} + z^63q^{12} + z^110q^{13} + z^93q^{14} + 5q^{15} + z^104q^{16} + z^110q^{17} + z^56q^{18} + z^28q^{19} + z^87q^{20} + z^13q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^104q^{2} + 5q^{3} + z^15q^{4} + 4q^{5} + z^32q^{6} + z^85q^{7} + z^38q^{8} + 3q^{9} + z^8q^{10} + z^41q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^79)\omega^{1}&0\\0&u(z^41)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 2187q^{3} + -(1173233073618776421 + O(11^20))*a + 170130902257686544534 + O(11^20)q^{4} + 78125q^{5} + 2187*aq^{6} + -(14078796883425318632 + O(11^20))*a + 23320842294559336013 + O(11^20)q^{7} + -(224087517061186304777 + O(11^20))*a + 203002574455249602762 + O(11^20)q^{8} + 4782969q^{9} + 78125*aq^{10} + (272190073079556157872*11 + O(11^21))*a + 221880377237764170119*11 + O(11^21)q^{11} + (125139247725976004077 + O(11^20))*a + 45536039854787807705 + O(11^20)q^{12} + -(126998270124211985643 + O(11^20))*a + 230983493093805052022 + O(11^20)q^{13} + -(162591953484007680519 + O(11^20))*a + 38203319611966739580 + O(11^20)q^{14} + 170859375q^{15} + -(1941196452709988715100 + O(11^21))*a + 18877802782488392113*11 + O(11^21)q^{16} + -(246406075522738024543 + O(11^20))*a + 20770961843675563051 + O(11^20)q^{17} + 4782969*aq^{18} + -(183763481661406868834 + O(11^20))*a - 97056046915720777000 + O(11^20)q^{19} + -(164834565638746639289 + O(11^20))*a - 44911000826810065407 + O(11^20)q^{20} + (156170982846588575062 + O(11^20))*a - 126317516673292838845 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 1173233073618776421 + O(11^20)x + -170130902257686577302 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 5q^{3} + 2q^{4} + 7q^{5} + 7q^{6} + 4q^{7} + 4q^{8} + 3q^{9} + q^{10} + 10q^{12} + q^{13} + 10q^{14} + 2q^{15} + 7q^{16} + 9q^{17} + 2q^{18} + 3q^{19} + 3q^{20} + 9q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 9q^{3} + 6q^{4} + 8q^{5} + 8q^{6} + q^{7} + 5q^{8} + 4q^{9} + q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(2)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 144091499507995928776 + O(11^20)q^{2} + 6561q^{3} + 109476704870136626651 + O(11^20)q^{4} + -390625q^{5} + 170585391714475771931 + O(11^20)q^{6} + 58429780546332325607 + O(11^20)q^{7} + 250147478686674133613 + O(11^20)q^{8} + 43046721q^{9} + -113669278276508323335 + O(11^20)q^{10} + -8007983966730741025*11 + O(11^20)q^{11} + -220333935007682369457 + O(11^20)q^{12} + -131611622561919349299 + O(11^20)q^{13} + -16950338279492379015 + O(11^20)q^{14} + -2562890625q^{15} + -124372889088372345832 + O(11^20)q^{16} + -296235042838219826662 + O(11^20)q^{17} + -245236529104315671173 + O(11^20)q^{18} + -311591404790892123675 + O(11^20)q^{19} + -311662014010240676109 + O(11^20)q^{20} + -109706947072816937043 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 5q^{3} + 2q^{4} + 7q^{5} + 4q^{6} + 7q^{7} + 7q^{8} + 3q^{9} + 10q^{10} + 10q^{12} + 10q^{13} + 10q^{14} + 2q^{15} + 7q^{16} + 2q^{17} + 9q^{18} + 8q^{19} + 3q^{20} + 2q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 9q^{3} + 6q^{4} + 8q^{5} + 8q^{6} + q^{7} + 5q^{8} + 4q^{9} + q^{10} + 2q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{6}&0\\0&u(2)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -83056145758732442722 + O(11^20)q^{2} + 6561q^{3} + 157074771589844060988 + O(11^20)q^{4} + -390625q^{5} + -3876427669949246232 + O(11^20)q^{6} + -186594992576883609294 + O(11^20)q^{7} + -320166393848196123529 + O(11^20)q^{8} + 43046721q^{9} + -234318612778565446176 + O(11^20)q^{10} + -16828793367704097809*11 + O(11^20)q^{11} + -85415835715049953664 + O(11^20)q^{12} + -315405969455155319610 + O(11^20)q^{13} + 56282909335615634495 + O(11^20)q^{14} + -2562890625q^{15} + 110475912834999708883 + O(11^20)q^{16} + 132013113992893347403 + O(11^20)q^{17} + 131257864900275821486 + O(11^20)q^{18} + -173945939567497880896 + O(11^20)q^{19} + 157885546366755730504 + O(11^20)q^{20} + 155244480325856167886 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 6q^{3} + 9q^{4} + 7q^{5} + 9q^{6} + 3q^{8} + 3q^{9} + 5q^{10} + 10q^{12} + 3q^{13} + 9q^{15} + 2q^{16} + 10q^{18} + 6q^{19} + 8q^{20} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 9q^{2} + 2q^{3} + 5q^{4} + 8q^{5} + 7q^{6} + 10q^{8} + 4q^{9} + 6q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{6}&0\\0&u(10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -238740975694266776999 + O(11^20)q^{2} + -6561q^{3} + 67697048275815856401 + O(11^20)q^{4} + -390625q^{5} + 217553327084622470511 + O(11^20)q^{6} + -24609630446542978132*11 + O(11^20)q^{7} + -100959907687354864409 + O(11^20)q^{8} + 43046721q^{9} + 243833031626169773353 + O(11^20)q^{10} + 29656704464430710225*11 + O(11^20)q^{11} + -145337082138227774301 + O(11^20)q^{12} + 64314536475298563688 + O(11^20)q^{13} + 7167329417656227442*11 + O(11^20)q^{14} + 2562890625q^{15} + 302628621484872093512 + O(11^20)q^{16} + 28620898847104368031*11 + O(11^20)q^{17} + 208110244684310501851 + O(11^20)q^{18} + 12300253256379420525 + O(11^20)q^{19} + 297318068499935032283 + O(11^20)q^{20} + 3786575954077315812*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 6q^{3} + 9q^{4} + 7q^{5} + 2q^{6} + 8q^{8} + 3q^{9} + 6q^{10} + 10q^{12} + 8q^{13} + 9q^{15} + 2q^{16} + q^{18} + 5q^{19} + 8q^{20} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 9q^{2} + 2q^{3} + 5q^{4} + 8q^{5} + 7q^{6} + 10q^{8} + 4q^{9} + 6q^{10} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(10)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -135805765055399654970 + O(11^20)q^{2} + -6561q^{3} + -148335122963298741054 + O(11^20)q^{4} + -390625q^{5} + 300631237767684076046 + O(11^20)q^{6} + 7004482980090190389*11 + O(11^20)q^{7} + -229869742498938163053 + O(11^20)q^{8} + 43046721q^{9} + 98874353403257120596 + O(11^20)q^{10} + -27269869694657921619*11 + O(11^20)q^{11} + -242500905211293258553 + O(11^20)q^{12} + -316840883562028248690 + O(11^20)q^{13} + -20522955797606941227*11 + O(11^20)q^{14} + 2562890625q^{15} + -50591760435654548933 + O(11^20)q^{16} + 355347975234023944*11 + O(11^20)q^{17} + 61434148490724039526 + O(11^20)q^{18} + 295147361900172477151 + O(11^20)q^{19} + 123093992109691745821 + O(11^20)q^{20} + -25935905612414877688*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^65q^{2} + 6q^{3} + z^73q^{4} + 4q^{5} + z^53q^{6} + z^38q^{7} + z^83q^{8} + 3q^{9} + z^89q^{10} + z^61q^{12} + z^45q^{13} + z^103q^{14} + 2q^{15} + z^101q^{16} + z^110q^{17} + z^41q^{18} + z^29q^{19} + z^97q^{20} + z^26q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^53q^{2} + 2q^{3} + z^49q^{4} + 3q^{5} + z^65q^{6} + z^74q^{7} + z^47q^{8} + 4q^{9} + z^29q^{10} + z^88q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^32)\omega^{6}&0\\0&u(z^88)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -6561q^{3} + -(294239901055521235729 + O(11^20))*a + 114822294975997915421 + O(11^20)q^{4} + 390625q^{5} + -6561*aq^{6} + -(182860995531218352206 + O(11^20))*a - 160315214313877611370 + O(11^20)q^{7} + (298581499405063963629 + O(11^20))*a - 175470314453330113483 + O(11^20)q^{8} + 43046721q^{9} + 390625*aq^{10} + -(30356078969905714178*11 + O(11^21))*a + 256005584773313217905*11 + O(11^21)q^{11} + -(284494631172398788901 + O(11^20))*a + 130916986944887227939 + O(11^20)q^{12} + -(244172570624505595298 + O(11^20))*a - 11795596608678715293 + O(11^20)q^{13} + -(21534421088057794978 + O(11^20))*a + 46665876610504753447 + O(11^20)q^{14} + -2562890625q^{15} + -(150051480140991475401 + O(11^20))*a - 19610294317155925594*11 + O(11^20)q^{16} + (33385607141272890121 + O(11^20))*a + 97740510554191706217 + O(11^20)q^{17} + 43046721*aq^{18} + -(1322015103737185895811 + O(11^21))*a + 7661160748071970522*11 + O(11^21)q^{19} + -(142965569902814677378 + O(11^20))*a + 216812845409897897455 + O(11^20)q^{20} + (237750715569112418183 + O(11^20))*a + 319879033759713817407 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 294239901055521235729 + O(11^20)x + -114822294975998046493 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^115q^{2} + 6q^{3} + z^83q^{4} + 4q^{5} + z^103q^{6} + z^58q^{7} + z^73q^{8} + 3q^{9} + z^19q^{10} + z^71q^{12} + z^15q^{13} + z^53q^{14} + 2q^{15} + z^31q^{16} + z^10q^{17} + z^91q^{18} + z^79q^{19} + z^107q^{20} + z^46q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^103q^{2} + 2q^{3} + z^59q^{4} + 3q^{5} + z^115q^{6} + z^94q^{7} + z^37q^{8} + 4q^{9} + z^79q^{10} + z^8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^112)\omega^{6}&0\\0&u(z^8)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -6561q^{3} + -(294239901055521235729 + O(11^20))*a + 114822294975997915421 + O(11^20)q^{4} + 390625q^{5} + -6561*aq^{6} + -(182860995531218352206 + O(11^20))*a - 160315214313877611370 + O(11^20)q^{7} + (298581499405063963629 + O(11^20))*a - 175470314453330113483 + O(11^20)q^{8} + 43046721q^{9} + 390625*aq^{10} + -(30356078969905714178*11 + O(11^21))*a + 256005584773313217905*11 + O(11^21)q^{11} + -(284494631172398788901 + O(11^20))*a + 130916986944887227939 + O(11^20)q^{12} + -(244172570624505595298 + O(11^20))*a - 11795596608678715293 + O(11^20)q^{13} + -(21534421088057794978 + O(11^20))*a + 46665876610504753447 + O(11^20)q^{14} + -2562890625q^{15} + -(150051480140991475401 + O(11^20))*a - 19610294317155925594*11 + O(11^20)q^{16} + (33385607141272890121 + O(11^20))*a + 97740510554191706217 + O(11^20)q^{17} + 43046721*aq^{18} + -(1322015103737185895811 + O(11^21))*a + 7661160748071970522*11 + O(11^21)q^{19} + -(142965569902814677378 + O(11^20))*a + 216812845409897897455 + O(11^20)q^{20} + (237750715569112418183 + O(11^20))*a + 319879033759713817407 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 294239901055521235729 + O(11^20)x + -114822294975998046493 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^5q^{2} + 6q^{3} + z^73q^{4} + 4q^{5} + z^113q^{6} + z^98q^{7} + z^23q^{8} + 3q^{9} + z^29q^{10} + z^61q^{12} + z^105q^{13} + z^103q^{14} + 2q^{15} + z^101q^{16} + z^50q^{17} + z^101q^{18} + z^89q^{19} + z^97q^{20} + z^86q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^53q^{2} + 2q^{3} + z^49q^{4} + 3q^{5} + z^65q^{6} + z^74q^{7} + z^47q^{8} + 4q^{9} + z^29q^{10} + z^88q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^32)\omega^{1}&0\\0&u(z^88)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -6561q^{3} + (294239901055521235696 + O(11^20))*a - 7164693628703225324 + O(11^20)q^{4} + 390625q^{5} + -6561*aq^{6} + (43589035088909358432 + O(11^20))*a + 33634958679821262189 + O(11^20)q^{7} + (266510894180201532688 + O(11^20))*a - 13480555344569036689 + O(11^20)q^{8} + 43046721q^{9} + 390625*aq^{10} + (38360630785935188218*11 + O(11^21))*a + 183347903536338807198*11 + O(11^21)q^{11} + (284494631172399005414 + O(11^20))*a - 84944747357339293306 + O(11^20)q^{12} + (165411008537564897871 + O(11^20))*a + 286924629454608078229 + O(11^20)q^{13} + (148214676722131727263 + O(11^20))*a + 114219855589201553036 + O(11^20)q^{14} + -2562890625q^{15} + -(333747644993601406631 + O(11^20))*a - 11422432062816364250*11 + O(11^20)q^{16} + -(208419025718548025544 + O(11^20))*a + 206972516718361955204 + O(11^20)q^{17} + 43046721*aq^{18} + -(3526835559388213305149 + O(11^21))*a + 305778424457468140583*11 + O(11^21)q^{19} + (142965569902801786753 + O(11^20))*a - 68469792747753911340 + O(11^20)q^{20} + -(68911371996296761927 + O(11^20))*a - 16965560427618204101 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -294239901055521235696 + O(11^20)x + 7164693628703094252 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^55q^{2} + 6q^{3} + z^83q^{4} + 4q^{5} + z^43q^{6} + z^118q^{7} + z^13q^{8} + 3q^{9} + z^79q^{10} + z^71q^{12} + z^75q^{13} + z^53q^{14} + 2q^{15} + z^31q^{16} + z^70q^{17} + z^31q^{18} + z^19q^{19} + z^107q^{20} + z^106q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + z^103q^{2} + 2q^{3} + z^59q^{4} + 3q^{5} + z^115q^{6} + z^94q^{7} + z^37q^{8} + 4q^{9} + z^79q^{10} + z^8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^112)\omega^{1}&0\\0&u(z^8)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -6561q^{3} + (294239901055521235696 + O(11^20))*a - 7164693628703225324 + O(11^20)q^{4} + 390625q^{5} + -6561*aq^{6} + (43589035088909358432 + O(11^20))*a + 33634958679821262189 + O(11^20)q^{7} + (266510894180201532688 + O(11^20))*a - 13480555344569036689 + O(11^20)q^{8} + 43046721q^{9} + 390625*aq^{10} + (38360630785935188218*11 + O(11^21))*a + 183347903536338807198*11 + O(11^21)q^{11} + (284494631172399005414 + O(11^20))*a - 84944747357339293306 + O(11^20)q^{12} + (165411008537564897871 + O(11^20))*a + 286924629454608078229 + O(11^20)q^{13} + (148214676722131727263 + O(11^20))*a + 114219855589201553036 + O(11^20)q^{14} + -2562890625q^{15} + -(333747644993601406631 + O(11^20))*a - 11422432062816364250*11 + O(11^20)q^{16} + -(208419025718548025544 + O(11^20))*a + 206972516718361955204 + O(11^20)q^{17} + 43046721*aq^{18} + -(3526835559388213305149 + O(11^21))*a + 305778424457468140583*11 + O(11^21)q^{19} + (142965569902801786753 + O(11^20))*a - 68469792747753911340 + O(11^20)q^{20} + -(68911371996296761927 + O(11^20))*a - 16965560427618204101 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -294239901055521235696 + O(11^20)x + 7164693628703094252 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 7q^{3} + 9q^{4} + 2q^{5} + 8q^{6} + 4q^{7} + 5q^{8} + 5q^{9} + 7q^{10} + 8q^{12} + 2q^{13} + 3q^{14} + 3q^{15} + 2q^{16} + 7q^{17} + q^{18} + 7q^{19} + 7q^{20} + 6q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 3q^{2} + 8q^{3} + q^{4} + 6q^{5} + 2q^{6} + 9q^{7} + q^{8} + 9q^{9} + 7q^{10} + 9q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(5)\omega^{1}&0\\0&u(9)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 136785601983731981234 + O(11^20)q^{2} + -19683q^{3} + 300595354336332175646 + O(11^20)q^{4} + -1953125q^{5} + -5524125691429806420 + O(11^20)q^{6} + 97665506940152702247 + O(11^20)q^{7} + -326623126542062680908 + O(11^20)q^{8} + 387420489q^{9} + -264636832957793801135 + O(11^20)q^{10} + -2925828859094444631*11 + O(11^20)q^{11} + 217846029839067682577 + O(11^20)q^{12} + -222451549851631958955 + O(11^20)q^{13} + -272487727606148055845 + O(11^20)q^{14} + 38443359375q^{15} + 254347926289653450410 + O(11^20)q^{16} + 183928672015313821845 + O(11^20)q^{17} + -254133194661841725702 + O(11^20)q^{18} + 316399609497009090466 + O(11^20)q^{19} + -126610437783809000663 + O(11^20)q^{20} + -303437580701692040444 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 7q^{3} + 10q^{4} + 2q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 5q^{9} + 3q^{10} + 4q^{12} + 6q^{13} + 3q^{14} + 3q^{15} + 4q^{16} + 8q^{17} + 2q^{18} + 9q^{19} + 9q^{20} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 9q^{2} + 6q^{3} + 8q^{4} + 7q^{5} + 10q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 8q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{8}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -256522122560367353837 + O(11^20)q^{2} + -19683q^{3} + -293809834342100467161 + O(11^20)q^{4} + -1953125q^{5} + 136226386847756520166 + O(11^20)q^{6} + 35320998511839791434 + O(11^20)q^{7} + 223275966655047534003 + O(11^20)q^{8} + 387420489q^{9} + -24100387657929406909 + O(11^20)q^{10} + 28239351567159364725*11 + O(11^20)q^{11} + 100012915277656038167 + O(11^20)q^{12} + -174186288794626801445 + O(11^20)q^{13} + 141171898553229274162 + O(11^20)q^{14} + 38443359375q^{15} + -323366018483208830301 + O(11^20)q^{16} + 102804194724677631435 + O(11^20)q^{17} + 237507476792610247808 + O(11^20)q^{18} + 140582060582945287733 + O(11^20)q^{19} + 332771875410355494738 + O(11^20)q^{20} + -272468943208125290789 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{3} + 5q^{4} + 2q^{5} + 8q^{7} + 5q^{9} + 9q^{12} + 10q^{13} + 8q^{15} + 3q^{16} + 10q^{17} + 5q^{19} + 10q^{20} + 10q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{3} + 4q^{4} + 7q^{5} + 9q^{7} + 3q^{9} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{8}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -13385944252755311599*11 + O(11^20)q^{2} + 19683q^{3} + 2091421336074949739481 + O(11^21)q^{4} + -1953125q^{5} + -2179075212932781489*11 + O(11^20)q^{6} + 28281943370054692271 + O(11^20)q^{7} + -8129216555991772275*11 + O(11^20)q^{8} + 387420489q^{9} + 12065593570889227613*11 + O(11^20)q^{10} + 30556705630075235650*11 + O(11^20)q^{11} + -2144281944908920832270 + O(11^21)q^{12} + -236753426854890540387 + O(11^20)q^{13} + 3264900363460145792*11 + O(11^20)q^{14} + -38443359375q^{15} + -25046761515325317083 + O(11^20)q^{16} + 131406425079796706784 + O(11^20)q^{17} + -18215011817341097996*11 + O(11^20)q^{18} + -55677164650094837845 + O(11^20)q^{19} + -2532289878480937177923 + O(11^21)q^{20} + 309245543559380360866 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^76q^{2} + 4q^{3} + z^69q^{4} + 9q^{5} + z^100q^{6} + z^59q^{7} + z^94q^{8} + 5q^{9} + z^28q^{10} + z^93q^{12} + z^94q^{13} + z^15q^{14} + 3q^{15} + z^40q^{16} + z^46q^{17} + z^4q^{18} + z^80q^{19} + z^21q^{20} + z^83q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^64q^{2} + 5q^{3} + z^45q^{4} + 4q^{5} + z^112q^{6} + z^95q^{7} + z^58q^{8} + 3q^{9} + z^88q^{10} + z^91q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^29)\omega^{8}&0\\0&u(z^91)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 19683q^{3} + -(222610656384762127963 + O(11^20))*a + 313721011611536908837 + O(11^20)q^{4} + 1953125q^{5} + 19683*aq^{6} + (64606395817342351997 + O(11^20))*a - 30942266912528557953 + O(11^20)q^{7} + (168187788115108491586 + O(11^20))*a + 181945891371709285937 + O(11^20)q^{8} + 387420489q^{9} + 1953125*aq^{10} + (21927269682871384572*11 + O(11^20))*a - 27901945986853852644*11 + O(11^20)q^{11} + -(24832625509624769616 + O(11^20))*a - 201531936087347817308 + O(11^20)q^{12} + -(20491099359732284273 + O(11^20))*a + 242848904165154107321 + O(11^20)q^{13} + (314725666747629700026 + O(11^20))*a + 98018528546516815802 + O(11^20)q^{14} + 38443359375q^{15} + -(312996449125848242924 + O(11^20))*a - 19626892139790663056*11 + O(11^20)q^{16} + (69199051019447779465 + O(11^20))*a - 26102353076075774059 + O(11^20)q^{17} + 387420489*aq^{18} + (69256629551252867002 + O(11^20))*a + 78999739937773680145 + O(11^20)q^{19} + -(226026483783311293693 + O(11^20))*a - 308080823035947949969 + O(11^20)q^{20} + (150198450211096967061 + O(11^20))*a - 197894225332797861994 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 222610656384762127963 + O(11^20)x + -313721011611537433125 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 3q^{4} + 9q^{5} + q^{6} + 4q^{7} + 2q^{8} + 5q^{9} + 5q^{10} + q^{12} + q^{14} + 3q^{15} + 10q^{16} + 10q^{17} + 4q^{18} + 10q^{19} + 5q^{20} + 5q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 3q^{6} + 9q^{7} + 7q^{8} + 9q^{9} + 5q^{10} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{1}&0\\0&u(8)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 26052737474758099680 + O(11^20)q^{2} + 19683q^{3} + -105970766635179373518 + O(11^20)q^{4} + 1953125q^{5} + 160535577052948990278 + O(11^20)q^{6} + 221620687952904688353 + O(11^20)q^{7} + -17269459981731541934 + O(11^20)q^{8} + 387420489q^{9} + 134263667804581573164 + O(11^20)q^{10} + -28120477431931020076*11 + O(11^20)q^{11} + -297615389299580431694 + O(11^20)q^{12} + 23134896465683320752*11 + O(11^20)q^{13} + 212520985605451783118 + O(11^20)q^{14} + 38443359375q^{15} + -203747445321801421096 + O(11^20)q^{16} + -161063351795525658568 + O(11^20)q^{17} + -84963065039387575223 + O(11^20)q^{18} + 254786177121018190911 + O(11^20)q^{19} + 73356647103168380704 + O(11^20)q^{20} + 49033834303881192815 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^116q^{2} + 4q^{3} + z^39q^{4} + 9q^{5} + z^20q^{6} + z^49q^{7} + z^74q^{8} + 5q^{9} + z^68q^{10} + z^63q^{12} + z^74q^{13} + z^45q^{14} + 3q^{15} + z^80q^{16} + z^26q^{17} + z^44q^{18} + z^40q^{19} + z^111q^{20} + z^73q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^104q^{2} + 5q^{3} + z^15q^{4} + 4q^{5} + z^32q^{6} + z^85q^{7} + z^38q^{8} + 3q^{9} + z^8q^{10} + z^41q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^79)\omega^{8}&0\\0&u(z^41)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 19683q^{3} + -(222610656384762127963 + O(11^20))*a + 313721011611536908837 + O(11^20)q^{4} + 1953125q^{5} + 19683*aq^{6} + (64606395817342351997 + O(11^20))*a - 30942266912528557953 + O(11^20)q^{7} + (168187788115108491586 + O(11^20))*a + 181945891371709285937 + O(11^20)q^{8} + 387420489q^{9} + 1953125*aq^{10} + (21927269682871384572*11 + O(11^20))*a - 27901945986853852644*11 + O(11^20)q^{11} + -(24832625509624769616 + O(11^20))*a - 201531936087347817308 + O(11^20)q^{12} + -(20491099359732284273 + O(11^20))*a + 242848904165154107321 + O(11^20)q^{13} + (314725666747629700026 + O(11^20))*a + 98018528546516815802 + O(11^20)q^{14} + 38443359375q^{15} + -(312996449125848242924 + O(11^20))*a - 19626892139790663056*11 + O(11^20)q^{16} + (69199051019447779465 + O(11^20))*a - 26102353076075774059 + O(11^20)q^{17} + 387420489*aq^{18} + (69256629551252867002 + O(11^20))*a + 78999739937773680145 + O(11^20)q^{19} + -(226026483783311293693 + O(11^20))*a - 308080823035947949969 + O(11^20)q^{20} + (150198450211096967061 + O(11^20))*a - 197894225332797861994 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 222610656384762127963 + O(11^20)x + -313721011611537433125 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{3} + 9q^{4} + 10q^{5} + 5q^{7} + q^{9} + 2q^{12} + q^{15} + 4q^{16} + q^{17} + 7q^{19} + 2q^{20} + 6q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{3} + 5q^{4} + 2q^{5} + 7q^{7} + 5q^{9} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{0}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -449696466023333965*11^2 + O(11^20)q^{2} + -59049q^{3} + 23947209566261751142756 + O(11^22)q^{4} + -9765625q^{5} + -38467027050253971*11^2 + O(11^20)q^{6} + -25461347783878707718 + O(11^20)q^{7} + 899034572903146433*11^2 + O(11^20)q^{8} + 3486784401q^{9} + -358060480190092497*11^2 + O(11^20)q^{10} + 29672750548738992867*11 + O(11^20)q^{11} + -11618079396652929099953 + O(11^22)q^{12} + -1867632963716818571*11^2 + O(11^20)q^{13} + 2079505566176968202*11^2 + O(11^20)q^{14} + 576650390625q^{15} + 316899002075757946024 + O(11^20)q^{16} + 76120333812466968740 + O(11^20)q^{17} + -2566700927875942150*11^2 + O(11^20)q^{18} + 240718489574801163691 + O(11^20)q^{19} + -33192303955833611174551 + O(11^22)q^{20} + -129113384017808524053 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^25q^{2} + q^{3} + z^69q^{4} + 10q^{5} + z^25q^{6} + z^107q^{7} + 5q^{8} + q^{9} + z^85q^{10} + z^69q^{12} + z^37q^{13} + 2q^{14} + 10q^{15} + z^39q^{16} + 6q^{17} + z^25q^{18} + 10q^{19} + z^9q^{20} + z^107q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^13q^{2} + 4q^{3} + z^45q^{4} + 2q^{5} + z^37q^{6} + z^23q^{7} + 2q^{8} + 5q^{9} + z^25q^{10} + z^55q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^65)\omega^{0}&0\\0&u(z^55)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 59049q^{3} + (142138691934458746246 + O(11^20))*a - 282172714833381759872 + O(11^20)q^{4} + -9765625q^{5} + 59049*aq^{6} + (523364949406308688329 + O(11^21))*a - 491097273432350022734 + O(11^21)q^{7} + (25897689747917138049*11^2 + O(11^21))*a + 2823058260794606093957 + O(11^21)q^{8} + 3486784401q^{9} + -9765625*aq^{10} + (19871364951828585302*11 + O(11^20))*a - 4697816821901356801*11 + O(11^20)q^{11} + -(81316740764167711622 + O(11^20))*a - 17513701645790800561 + O(11^20)q^{12} + (1583649235420496908864 + O(11^21))*a - 4586203085723360785*11^2 + O(11^21)q^{13} + (2647011600635843899*11^2 + O(11^20))*a + 266753617092379466920 + O(11^20)q^{14} + -576650390625q^{15} + -(282143660104994842115 + O(11^20))*a + 67855275535540325538 + O(11^20)q^{16} + (1173271191052172353*11^2 + O(11^20))*a - 114501784419210639896 + O(11^20)q^{17} + 3486784401*aq^{18} + -(1940809024862119702*11^2 + O(11^20))*a - 158868869932209905480 + O(11^20)q^{19} + -(208378006412904336068 + O(11^20))*a + 193301043827782584387 + O(11^20)q^{20} + (733130271045154481985 + O(11^21))*a + 2776632640892944225943 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -142138691934458746246 + O(11^20)x + 282172714833379662720 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^35q^{2} + q^{3} + z^39q^{4} + 10q^{5} + z^35q^{6} + z^97q^{7} + 5q^{8} + q^{9} + z^95q^{10} + z^39q^{12} + z^47q^{13} + 2q^{14} + 10q^{15} + z^69q^{16} + 6q^{17} + z^35q^{18} + 10q^{19} + z^99q^{20} + z^97q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^23q^{2} + 4q^{3} + z^15q^{4} + 2q^{5} + z^47q^{6} + z^13q^{7} + 2q^{8} + 5q^{9} + z^35q^{10} + z^5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^115)\omega^{0}&0\\0&u(z^5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + 59049q^{3} + (142138691934458746246 + O(11^20))*a - 282172714833381759872 + O(11^20)q^{4} + -9765625q^{5} + 59049*aq^{6} + (523364949406308688329 + O(11^21))*a - 491097273432350022734 + O(11^21)q^{7} + (25897689747917138049*11^2 + O(11^21))*a + 2823058260794606093957 + O(11^21)q^{8} + 3486784401q^{9} + -9765625*aq^{10} + (19871364951828585302*11 + O(11^20))*a - 4697816821901356801*11 + O(11^20)q^{11} + -(81316740764167711622 + O(11^20))*a - 17513701645790800561 + O(11^20)q^{12} + (1583649235420496908864 + O(11^21))*a - 4586203085723360785*11^2 + O(11^21)q^{13} + (2647011600635843899*11^2 + O(11^20))*a + 266753617092379466920 + O(11^20)q^{14} + -576650390625q^{15} + -(282143660104994842115 + O(11^20))*a + 67855275535540325538 + O(11^20)q^{16} + (1173271191052172353*11^2 + O(11^20))*a - 114501784419210639896 + O(11^20)q^{17} + 3486784401*aq^{18} + -(1940809024862119702*11^2 + O(11^20))*a - 158868869932209905480 + O(11^20)q^{19} + -(208378006412904336068 + O(11^20))*a + 193301043827782584387 + O(11^20)q^{20} + (733130271045154481985 + O(11^21))*a + 2776632640892944225943 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -142138691934458746246 + O(11^20)x + 282172714833379662720 + O(11^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 10q^{3} + 10q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 10q^{10} + q^{12} + 9q^{13} + 10q^{15} + 10q^{16} + 2q^{17} + 10q^{18} + 4q^{19} + 10q^{20} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$2$ newform
$g = q + 10q^{2} + 10q^{3} + 10q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 10q^{10} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{0}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 211952552600926859490 + O(11^20)q^{2} + -59049q^{3} + 71922571603221830886 + O(11^20)q^{4} + 9765625q^{5} + 254627193216285150394 + O(11^20)q^{6} + -26641012234877748969*11 + O(11^20)q^{7} + 187250963787395858342 + O(11^20)q^{8} + 3486784401q^{9} + -90165586085541676249 + O(11^20)q^{10} + 1110088387966295818*11 + O(11^19)q^{11} + 114787410605446098499 + O(11^20)q^{12} + -223338416819695273032 + O(11^20)q^{13} + 20464147735831640255*11 + O(11^20)q^{14} + -576650390625q^{15} + 112119154895973352434 + O(11^20)q^{16} + -225579095873058280219 + O(11^20)q^{17} + -191495480638199982157 + O(11^20)q^{18} + -117257756067506383644 + O(11^20)q^{19} + -295646742636479998677 + O(11^20)q^{20} + -8993056822760826621*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^61q^{2} + 10q^{3} + z^79q^{4} + q^{5} + zq^{6} + z^63q^{7} + z^65q^{8} + q^{9} + z^61q^{10} + z^19q^{12} + z^73q^{13} + z^4q^{14} + 10q^{15} + z^110q^{16} + z^71q^{17} + z^61q^{18} + z^87q^{19} + z^79q^{20} + z^3q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^49q^{2} + 7q^{3} + z^55q^{4} + 9q^{5} + z^13q^{6} + z^99q^{7} + z^29q^{8} + 5q^{9} + zq^{10} + z^44q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^76)\omega^{0}&0\\0&u(z^44)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -59049q^{3} + -(116531532108928555337 + O(11^20))*a + 180709794610884586494 + O(11^20)q^{4} + 9765625q^{5} + -59049*aq^{6} + -(81977712067328285062 + O(11^20))*a + 71042637830402397156 + O(11^20)q^{7} + (295550882087195481669 + O(11^20))*a + 329256802437333451480 + O(11^20)q^{8} + 3486784401q^{9} + 9765625*aq^{10} + -(1983659416617462016*11 + O(11^19))*a - 1654147170006205599*11 + O(11^19)q^{11} + (183491329898489986685 + O(11^20))*a - 244992352789641947145 + O(11^20)q^{12} + (91123167035900883336 + O(11^20))*a - 1441280425060559551*11^2 + O(11^20)q^{13} + (266796386610660125413 + O(11^20))*a + 318332889382363925866 + O(11^20)q^{14} + -576650390625q^{15} + -(205943630303816993381 + O(11^20))*a - 222020281969157405306 + O(11^20)q^{16} + -(22027375094205645099 + O(11^20))*a + 267067300110921826567 + O(11^20)q^{17} + 3486784401*aq^{18} + (68672191816808670520 + O(11^20))*a + 161768551049321168270 + O(11^20)q^{19} + -(207073179821008764256 + O(11^20))*a - 243710272974955410430 + O(11^20)q^{20} + (265706323898638424843 + O(11^20))*a + 272247152013067712792 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 116531532108928555337 + O(11^20)x + -180709794610886683646 + O(11^20)$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 3q^{6} + 9q^{7} + 7q^{8} + 9q^{9} + 5q^{10} + 8q^{11} + q^{12} + 9q^{14} + 4q^{15} + 8q^{16} + 8q^{17} + 9q^{18} + 2q^{19} + 9q^{20} + 5q^{21} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 8q^{3} + q^{4} + 6q^{5} + 2q^{6} + 9q^{7} + q^{8} + 9q^{9} + 7q^{10} + 9q^{11} + 8q^{12} + 8q^{13} + 5q^{14} + 4q^{15} + 6q^{16} + 10q^{17} + 5q^{18} + 8q^{19} + 6q^{20} + 6q^{21} + \cdots \in S_{14}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 9q^{3} + 6q^{4} + 8q^{5} + 8q^{6} + q^{7} + 5q^{8} + 4q^{9} + q^{10} + 2q^{11} + 10q^{12} + 5q^{13} + 7q^{14} + 6q^{15} + 8q^{16} + 4q^{17} + 6q^{18} + q^{19} + 4q^{20} + 9q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^103q^{2} + 2q^{3} + z^59q^{4} + 3q^{5} + z^115q^{6} + z^94q^{7} + z^37q^{8} + 4q^{9} + z^79q^{10} + z^8q^{11} + z^71q^{12} + z^3q^{13} + z^77q^{14} + 6q^{15} + z^103q^{16} + z^22q^{17} + z^7q^{18} + z^43q^{19} + z^35q^{20} + z^106q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^53q^{2} + 2q^{3} + z^49q^{4} + 3q^{5} + z^65q^{6} + z^74q^{7} + z^47q^{8} + 4q^{9} + z^29q^{10} + z^88q^{11} + z^61q^{12} + z^33q^{13} + z^7q^{14} + 6q^{15} + z^53q^{16} + z^2q^{17} + z^77q^{18} + z^113q^{19} + z^25q^{20} + z^86q^{21} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 9q^{2} + 2q^{3} + 5q^{4} + 8q^{5} + 7q^{6} + 10q^{8} + 4q^{9} + 6q^{10} + 10q^{11} + 10q^{12} + 7q^{13} + 5q^{15} + 7q^{16} + 3q^{18} + 9q^{19} + 7q^{20} + \cdots \in S_{16}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^64q^{2} + 5q^{3} + z^45q^{4} + 4q^{5} + z^112q^{6} + z^95q^{7} + z^58q^{8} + 3q^{9} + z^88q^{10} + z^91q^{11} + z^93q^{12} + z^82q^{13} + z^39q^{14} + 9q^{15} + z^112q^{16} + z^58q^{17} + z^40q^{18} + z^44q^{19} + z^69q^{20} + z^23q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^104q^{2} + 5q^{3} + z^15q^{4} + 4q^{5} + z^32q^{6} + z^85q^{7} + z^38q^{8} + 3q^{9} + z^8q^{10} + z^41q^{11} + z^63q^{12} + z^62q^{13} + z^69q^{14} + 9q^{15} + z^32q^{16} + z^38q^{17} + z^80q^{18} + z^4q^{19} + z^39q^{20} + z^13q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{3} + 4q^{4} + 7q^{5} + 9q^{7} + 3q^{9} + 4q^{11} + 9q^{12} + 5q^{13} + 2q^{15} + 5q^{16} + 9q^{17} + 2q^{19} + 6q^{20} + q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 9q^{2} + 6q^{3} + 8q^{4} + 7q^{5} + 10q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 8q^{10} + 7q^{11} + 4q^{12} + 3q^{13} + q^{14} + 9q^{15} + 3q^{16} + 5q^{17} + 5q^{18} + 8q^{19} + q^{20} + 8q^{21} + \cdots \in S_{18}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^49q^{2} + 7q^{3} + z^55q^{4} + 9q^{5} + z^13q^{6} + z^99q^{7} + z^29q^{8} + 5q^{9} + zq^{10} + z^44q^{11} + z^19q^{12} + z^61q^{13} + z^28q^{14} + 8q^{15} + z^62q^{16} + z^83q^{17} + z^97q^{18} + z^51q^{19} + z^7q^{20} + z^63q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^59q^{2} + 7q^{3} + z^5q^{4} + 9q^{5} + z^23q^{6} + z^9q^{7} + z^79q^{8} + 5q^{9} + z^11q^{10} + z^4q^{11} + z^89q^{12} + z^71q^{13} + z^68q^{14} + 8q^{15} + z^82q^{16} + z^73q^{17} + z^107q^{18} + z^81q^{19} + z^77q^{20} + z^93q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{3} + 5q^{4} + 2q^{5} + 7q^{7} + 5q^{9} + 4q^{11} + 2q^{12} + 3q^{15} + 3q^{16} + 2q^{17} + 5q^{19} + 10q^{20} + 5q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^13q^{2} + 4q^{3} + z^45q^{4} + 2q^{5} + z^37q^{6} + z^23q^{7} + 2q^{8} + 5q^{9} + z^25q^{10} + z^55q^{11} + z^69q^{12} + z^25q^{13} + 8q^{14} + 8q^{15} + z^111q^{16} + q^{17} + z^61q^{18} + 4q^{19} + z^57q^{20} + z^47q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^23q^{2} + 4q^{3} + z^15q^{4} + 2q^{5} + z^47q^{6} + z^13q^{7} + 2q^{8} + 5q^{9} + z^35q^{10} + z^5q^{11} + z^39q^{12} + z^35q^{13} + 8q^{14} + 8q^{15} + z^21q^{16} + q^{17} + z^71q^{18} + 4q^{19} + z^27q^{20} + z^37q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 4q^{3} + 10q^{4} + 9q^{5} + 6q^{6} + 9q^{7} + 6q^{8} + 5q^{9} + 8q^{10} + q^{11} + 7q^{12} + 7q^{13} + 8q^{14} + 3q^{15} + 4q^{16} + 3q^{17} + 2q^{18} + 8q^{19} + 2q^{20} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + q^{3} + q^{4} + 10q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + q^{9} + 6q^{10} + 10q^{11} + q^{12} + 3q^{13} + 10q^{14} + 10q^{15} + 6q^{16} + 5q^{18} + 3q^{19} + 10q^{20} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^65q^{2} + 10q^{3} + z^32q^{4} + 10q^{5} + z^5q^{6} + z^7q^{7} + z^117q^{8} + q^{9} + z^5q^{10} + 10q^{11} + z^92q^{12} + z^90q^{13} + 9q^{14} + q^{15} + 3q^{16} + z^73q^{17} + z^65q^{18} + z^83q^{19} + z^92q^{20} + z^67q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^115q^{2} + 10q^{3} + z^112q^{4} + 10q^{5} + z^55q^{6} + z^77q^{7} + z^87q^{8} + q^{9} + z^55q^{10} + 10q^{11} + z^52q^{12} + z^30q^{13} + 9q^{14} + q^{15} + 3q^{16} + z^83q^{17} + z^115q^{18} + z^73q^{19} + z^52q^{20} + z^17q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + q^{3} + q^{4} + 10q^{5} + 6q^{6} + 2q^{7} + 5q^{8} + q^{9} + 5q^{10} + 10q^{11} + q^{12} + q^{13} + q^{14} + 10q^{15} + 6q^{16} + 6q^{18} + q^{19} + 10q^{20} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^880q^{2} + q^{3} + z^294q^{4} + q^{5} + z^880q^{6} + z^279q^{7} + z^434q^{8} + q^{9} + z^880q^{10} + q^{11} + z^294q^{12} + z^1203q^{13} + z^1159q^{14} + q^{15} + z^340q^{16} + z^403q^{17} + z^880q^{18} + z^194q^{19} + z^294q^{20} + z^279q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^80q^{2} + q^{3} + z^994q^{4} + q^{5} + z^80q^{6} + z^509q^{7} + z^644q^{8} + q^{9} + z^80q^{10} + q^{11} + z^994q^{12} + z^593q^{13} + z^589q^{14} + q^{15} + z^1240q^{16} + z^883q^{17} + z^80q^{18} + z^864q^{19} + z^994q^{20} + z^509q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^370q^{2} + q^{3} + z^574q^{4} + q^{5} + z^370q^{6} + z^409q^{7} + z^784q^{8} + q^{9} + z^370q^{10} + q^{11} + z^574q^{12} + z^1263q^{13} + z^779q^{14} + q^{15} + z^1080q^{16} + z^443q^{17} + z^370q^{18} + z^804q^{19} + z^574q^{20} + z^409q^{21} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 9=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + 10q^{3} + 10q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 10q^{10} + 7q^{11} + q^{12} + 9q^{13} + 10q^{15} + 10q^{16} + 2q^{17} + 10q^{18} + 4q^{19} + 10q^{20} + \cdots \in S_{22}(\Gamma_0(15);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 16, \quad \ell = 11}$ \quad (37 forms)}

\begin{enumerate}
\item  Consider
$f = q + q^{3} + 4q^{5} + 6q^{7} + 7q^{9} + 4q^{13} + 4q^{15} + 10q^{17} + 8q^{19} + 6q^{21} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{3} + 5q^{5} + 10q^{7} + 10q^{9} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(10)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 12q^{3} + -4330q^{5} + 139992q^{7} + -1594179q^{9} + 589484*11q^{11} + -22588034q^{13} + -51960q^{15} + -23732270q^{17} + -325344836q^{19} + 1679904q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{3} + 4q^{5} + 4q^{7} + 9q^{9} + 8q^{13} + 9q^{15} + 2q^{17} + 8q^{19} + 9q^{21} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{3} + 5q^{5} + 3q^{7} + 5q^{9} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{1}&0\\0&u(5)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -468q^{3} + 56214q^{5} + -333032q^{7} + -1375299q^{9} + 581580*11q^{11} + 15199742q^{13} + -26308152q^{15} + 43114194q^{17} + 365115484q^{19} + 155858976q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{3} + 3q^{5} + 8q^{7} + 5q^{13} + 10q^{15} + 9q^{17} + 9q^{19} + q^{21} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 2q^{3} + q^{5} + 6q^{7} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{1}&0\\0&u(4)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1236q^{3} + -57450q^{5} + -64232q^{7} + -6057*11q^{9} + -224052*11q^{11} + 8032766q^{13} + 71008200q^{15} + 71112402q^{17} + -136337060q^{19} + 79390752q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{3} + 9q^{5} + 9q^{7} + 3q^{15} + 6q^{17} + 3q^{21} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 9q^{5} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{3}$. The form
$f_0 = q + aq^{3} + -12*a + 4006q^{5} + -222*a - 152256q^{7} + -872*a + 1414557q^{9} + 531*a - 8005504q^{11} + 10452*a + 13929358q^{13} + 14470*a - 36106560q^{15} + -38184*a - 93545038q^{17} + -8529*11*a + 1678976*11q^{19} + 41328*a - 667971360q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 872x + -3008880$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 6q^{3} + 10q^{5} + 4q^{7} + 2q^{9} + 10q^{13} + 5q^{15} + 3q^{17} + 4q^{19} + 2q^{21} + \cdots \in S_{16}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 7q^{3} + 6q^{5} + 5q^{7} + 7q^{9} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{1}&0\\0&u(5)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2700q^{3} + -251890q^{5} + -1374072q^{7} + -7058907q^{9} + 3935156*11q^{11} + -323161466q^{13} + 680103000q^{15} + -191653646q^{17} + 6515456644q^{19} + 3709994400q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{3} + q^{5} + 8q^{7} + q^{15} + 3q^{17} + 8q^{21} + \cdots \in S_{16}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 9q^{5} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{15}$. The form
$f_0 = q + 3444q^{3} + 313358q^{5} + 2324616q^{7} + -226161*11q^{9} + 456604*11^2q^{11} + -10023598*11q^{13} + 1079204952q^{15} + -2601428750q^{17} + -177465844*11q^{19} + 8005977504q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 7q^{3} + 2q^{5} + 10q^{7} + 4q^{9} + 2q^{13} + 3q^{15} + 2q^{17} + 7q^{19} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 10q^{3} + 10q^{5} + 7q^{7} + 3q^{9} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{1}&0\\0&u(8)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -6252q^{3} + 90510q^{5} + -56q^{7} + 24738597q^{9} + 8717268*11q^{11} + -59782138q^{13} + -565868520q^{15} + -1355814414q^{17} + -3783593180q^{19} + 350112q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{5} + 9q^{7} + 10q^{9} + 10q^{13} + 2q^{17} + 3q^{19} + \cdots \in S_{16}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + q^{5} + 3q^{7} + 2q^{9} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(7)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -29659132878511889008*11 + O(11^20)q^{3} + 308266706037187893917 + O(11^20)q^{5} + 238458227305870040588 + O(11^20)q^{7} + 189360097501490988236 + O(11^20)q^{9} + -2366594861980184581*11 + O(11^20)q^{11} + -145163955559332709737 + O(11^20)q^{13} + -29743906027638592201*11 + O(11^20)q^{15} + -96112956299898508947 + O(11^20)q^{17} + -3473901814486932800936 + O(11^21)q^{19} + 6385330492892579432*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{3} + 4q^{5} + 8q^{7} + 5q^{9} + q^{13} + 2q^{15} + 5q^{17} + 9q^{19} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{1}&0\\0&u(8)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -237853258544855010238 + O(11^20)q^{3} + 143445619107540422809 + O(11^20)q^{5} + 151011389797562871838 + O(11^20)q^{7} + 280446157570764627678 + O(11^20)q^{9} + 6128292804387041005*11 + O(11^20)q^{11} + 136748187992773101625 + O(11^20)q^{13} + 181024828818751938484 + O(11^20)q^{15} + -335098537127383232363 + O(11^20)q^{17} + -779378992030693016008 + O(11^21)q^{19} + 306402192853309311070 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{3} + 6q^{5} + 7q^{15} + \cdots \in S_{18}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{3} + 10q^{5} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(10)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{3} + 36*a + 196182q^{5} + 54*11*a - 1311040*11q^{7} + 5880*a + 207300141q^{9} + 315*11^2*a - 67443840*11q^{11} + 14796*11*a - 9409330*11^2q^{13} + 407862*a + 12111850944q^{15} + 15912*11*a - 1296700650*11q^{17} + -260145*11*a - 3832157056*11q^{19} + -90320*11^2*a + 18167776416*11q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -5880x + -336440304$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{3} + 4q^{5} + 3q^{7} + 5q^{9} + 10q^{13} + 2q^{15} + 6q^{17} + 2q^{19} + 7q^{21} + \cdots \in S_{18}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{6}&0\\0&u(8)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 27120212785074008683 + O(11^20)q^{3} + -174075515412647438993 + O(11^20)q^{5} + 151823049696759149382 + O(11^20)q^{7} + -203258971072395800392 + O(11^20)q^{9} + 1912073832274955453*11 + O(11^20)q^{11} + 318101605297731572008 + O(11^20)q^{13} + -321580757079223955300 + O(11^20)q^{15} + 269735114044000647664 + O(11^20)q^{17} + -220005465942469156519 + O(11^20)q^{19} + -173162390089023029182 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{3} + 6q^{5} + 5q^{7} + 5q^{9} + 3q^{13} + 4q^{15} + 5q^{17} + 3q^{19} + 7q^{21} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 10q^{3} + 10q^{5} + 7q^{7} + 3q^{9} + 8q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(7)\omega^{8}&0\\0&u(8)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 53028q^{3} + -5556930q^{5} + 44496424q^{7} + 1649707317q^{9} + -574606812*11q^{11} + -33124973098q^{13} + -294672884040q^{15} + -722355252174q^{17} + 1312620671860q^{19} + 2359556371872q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{5} + 10q^{7} + 7q^{9} + 4q^{13} + 5q^{17} + 6q^{19} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + q^{5} + 3q^{7} + 2q^{9} + 7q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{8}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -28762896031311307819*11 + O(11^20)q^{3} + 879258209361647217272 + O(11^21)q^{5} + 2856029319604017868398 + O(11^21)q^{7} + 59689194141760551716 + O(11^20)q^{9} + 17682337531174718495*11 + O(11^20)q^{11} + -292846419252694870801 + O(11^20)q^{13} + -30133086800404770211*11 + O(11^20)q^{15} + 88133675594113824405 + O(11^20)q^{17} + 246354975700501317689 + O(11^20)q^{19} + 8799218661165635436*11 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{3} + 8q^{5} + 2q^{7} + 8q^{9} + 4q^{13} + 3q^{15} + 2q^{17} + 8q^{19} + 9q^{21} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 7q^{3} + 6q^{5} + 5q^{7} + 7q^{9} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{8}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -3012953843206747342 + O(11^20)q^{3} + 237341441956642283587 + O(11^20)q^{5} + -272316958307207048365 + O(11^20)q^{7} + 151616048286617080251 + O(11^20)q^{9} + -13055498409929419156*11 + O(11^20)q^{11} + 293985161745674099474 + O(11^20)q^{13} + 6080281819353922040 + O(11^20)q^{15} + -106390654215567499567 + O(11^20)q^{17} + -133460784114714809888 + O(11^20)q^{19} + 241850928540593178288 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{3} + 3q^{5} + 4q^{7} + 5q^{15} + 2q^{17} + 3q^{21} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 9q^{5} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{99}$. The form
$f_0 = q + aq^{3} + -(71739913100518356081 + O(11^20))*a - 286771297198932320206 + O(11^20)q^{5} + -(89298272145247099096 + O(11^20))*a + 319324010414393401097 + O(11^20)q^{7} + (3012953843206723610 + O(11^20))*a + 40305095844010507117 + O(11^20)q^{9} + -(261868908749327877655 + O(11^20))*a - 306917766934413219031 + O(11^20)q^{11} + (146960528936824745075 + O(11^20))*a + 283648462680551198426 + O(11^20)q^{13} + (49429855242292176837 + O(11^21))*a - 2114529052840512092740 + O(11^21)q^{15} + (5222660501278696115 + O(11^20))*a - 183128850258256009838 + O(11^20)q^{17} + (9252652515355046232*11 + O(11^20))*a - 526357107575210889*11^2 + O(11^20)q^{19} + -(47007052107242204452 + O(11^20))*a + 61024949400868555464 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3012953843206723610 + O(11^20)x + -40305095845172768584 + O(11^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 6q^{3} + 5q^{5} + 9q^{7} + 8q^{13} + 8q^{15} + 7q^{17} + 8q^{19} + 10q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 2q^{3} + q^{5} + 6q^{7} + 4q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(3)\omega^{0}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -71604q^{3} + -28693770q^{5} + 853202392q^{7} + -484838217*11q^{9} + -7884652692*11q^{11} + -895323442786q^{13} + 2054588707080q^{15} + 3257566804818q^{17} + -23032467644420q^{19} + -61092704076768q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{3} + 3q^{5} + 10q^{7} + q^{9} + 4q^{13} + 5q^{15} + 4q^{17} + q^{19} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{3} + 5q^{5} + 3q^{7} + 5q^{9} + 5q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(9)\omega^{0}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 197815801794667479828 + O(11^20)q^{3} + 10576129841434816530 + O(11^20)q^{5} + 186431864848526734843 + O(11^20)q^{7} + 118422391371438141362 + O(11^20)q^{9} + 24976012745428922040*11 + O(11^20)q^{11} + 298761524669349109867 + O(11^20)q^{13} + 310158248916494961548 + O(11^20)q^{15} + 66120954489554434749 + O(11^20)q^{17} + -39625584201932371499 + O(11^20)q^{19} + -3351172969707456781248 + O(11^21)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{3} + 4q^{5} + 6q^{7} + 9q^{15} + q^{17} + 8q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
We have $\rho_f \otimes \chi^{1} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 4q^{3} + 9q^{5} + 9q^{7} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{111}$. The form
$f_0 = q + -192011347013165958526 + O(11^20)q^{3} + 196273029332040884436 + O(11^20)q^{5} + 8163195851865990623 + O(11^20)q^{7} + 22955383498403253067*11 + O(11^20)q^{9} + 650348885411941718*11^2 + O(11^20)q^{11} + -117066435239381991577*11 + O(11^21)q^{13} + -94228476522828356579 + O(11^20)q^{15} + -162433658798202972476 + O(11^20)q^{17} + -26058816584734320422*11 + O(11^20)q^{19} + -235635184637186998876 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 4q^{3} + 3q^{5} + 4q^{7} + 2q^{9} + 2q^{13} + q^{15} + 9q^{17} + q^{19} + 5q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
We have $\rho_f \otimes \chi^{9} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{3} + 5q^{5} + 10q^{7} + 10q^{9} + 10q^{11} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{0}&0\\0&u(10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 54482294323998791782 + O(11^20)q^{3} + 325033101411364126993 + O(11^20)q^{5} + 36661258535087029210 + O(11^20)q^{7} + 165475433795445166417 + O(11^20)q^{9} + -16179817711094748369*11 + O(11^20)q^{11} + -322845341485774539144 + O(11^20)q^{13} + 329062144376499511957 + O(11^20)q^{15} + 267123841190985321431 + O(11^20)q^{17} + -70862335466729695272 + O(11^20)q^{19} + 180401648425834546243 + O(11^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 10q^{3} + 8q^{5} + 10q^{7} + 7q^{9} + 6q^{13} + 3q^{15} + 9q^{17} + 5q^{19} + q^{21} + \cdots \in S_{14}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{3} + 10q^{5} + 10q^{11} + 10q^{15} + \cdots \in S_{16}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{3} + 3q^{5} + q^{7} + 4q^{9} + 7q^{13} + q^{15} + 4q^{17} + 10q^{19} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 2q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + 8q^{11} + 5q^{13} + 6q^{15} + q^{17} + 3q^{19} + 4q^{21} + \cdots \in S_{16}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{3} + 10q^{5} + 5q^{9} + 6q^{15} + \cdots \in S_{18}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 10q^{3} + 10q^{5} + 7q^{7} + 3q^{9} + 8q^{11} + 7q^{13} + q^{15} + 10q^{17} + 10q^{19} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{5} + 3q^{7} + 2q^{9} + 7q^{11} + 2q^{13} + 10q^{17} + 9q^{19} + \cdots \in S_{18}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{3} + 6q^{5} + 5q^{7} + 7q^{9} + 5q^{11} + 2q^{13} + 9q^{15} + 4q^{17} + q^{19} + 2q^{21} + \cdots \in S_{18}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{3} + 5q^{5} + 3q^{7} + 5q^{9} + 5q^{11} + 2q^{13} + 4q^{15} + 8q^{17} + 7q^{19} + 9q^{21} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{3} + q^{5} + 6q^{7} + 4q^{11} + 4q^{13} + 2q^{15} + 3q^{17} + q^{19} + q^{21} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{3} + 9q^{5} + 6q^{7} + 5q^{9} + 5q^{13} + 5q^{15} + 10q^{17} + 9q^{19} + 7q^{21} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 5q^{3} + 5q^{5} + 10q^{7} + 10q^{9} + 10q^{11} + q^{13} + 3q^{15} + 7q^{17} + 7q^{19} + 6q^{21} + \cdots \in S_{20}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{3} + 6q^{5} + 3q^{7} + 2q^{9} + 3q^{13} + 9q^{15} + 7q^{17} + 2q^{19} + 10q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + q^{3} + q^{5} + 2q^{7} + 9q^{9} + 10q^{11} + 4q^{13} + q^{15} + 9q^{17} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{3} + 8q^{5} + 9q^{7} + 9q^{9} + q^{11} + 7q^{13} + 3q^{15} + 6q^{17} + 3q^{19} + 2q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{3} + 8q^{5} + 2q^{7} + 6q^{9} + q^{11} + 2q^{15} + 5q^{17} + 7q^{19} + 6q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^37q^{3} + z^116q^{5} + z^109q^{7} + z^76q^{9} + q^{11} + z^59q^{13} + z^33q^{15} + 2q^{17} + 4q^{19} + z^26q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^47q^{3} + z^76q^{5} + z^119q^{7} + z^116q^{9} + q^{11} + z^49q^{13} + z^3q^{15} + 2q^{17} + 4q^{19} + z^46q^{21} + \cdots \in S_{22}(\Gamma_0(16);\overline{\mathbf{F}}_{11}).$
Also,~$z$ satisfies the equation $x^{2} + 7x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\setcounter{section}{12}
\section{$\mathbf{2 \leq N \leq 10, \quad \ell = 13}$}

\subsection{$\mathbf{N = 2, \quad \ell = 13}$ \quad (10 forms)}

\begin{enumerate}
\item  Consider
$f = q + 9q^{2} + 3q^{4} + 12q^{5} + 12q^{7} + q^{8} + 4q^{9} + 4q^{10} + 10q^{11} + 4q^{14} + 9q^{16} + 9q^{17} + 10q^{18} + \cdots \in S_{18}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 9q^{4} + 12q^{5} + 10q^{7} + q^{8} + 12q^{9} + 10q^{10} + 12q^{11} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(7)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 256q^{2} + 36*13^2q^{3} + 65536q^{4} + 1255110q^{5} + 9216*13^2q^{6} + -22465912q^{7} + 16777216q^{8} + -92125107q^{9} + 321308160q^{10} + 172399692q^{11} + 2359296*13^2q^{12} + -167703802*13q^{13} + -5751273472q^{14} + 45183960*13^2q^{15} + 4294967296q^{16} + 30163933458q^{17} + -23584027392q^{18} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 12q^{3} + 12q^{4} + 2q^{5} + 5q^{6} + 11q^{7} + 5q^{8} + 11q^{9} + 3q^{10} + q^{12} + 10q^{14} + 11q^{15} + q^{16} + 3q^{17} + 10q^{18} + \cdots \in S_{20}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{2} + 12q^{3} + 12q^{4} + 11q^{5} + 8q^{6} + 2q^{7} + 8q^{8} + 11q^{9} + 3q^{10} + q^{12} + 4q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{1}&0\\0&u(4)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -512q^{2} + -13092q^{3} + 262144q^{4} + 6546750q^{5} + 6703104q^{6} + 96674264q^{7} + -134217728q^{8} + -990861003q^{9} + -3351936000q^{10} + 907668804*13q^{11} + -3431989248q^{12} + 2646286766*13q^{13} + -49497223168q^{14} + -85710051000q^{15} + 68719476736q^{16} + -400697609166q^{17} + 507320833536q^{18} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 10q^{3} + 9q^{4} + 3q^{5} + 9q^{6} + q^{7} + 12q^{8} + 8q^{9} + 4q^{10} + 12q^{12} + 10q^{14} + 4q^{15} + 3q^{16} + 12q^{17} + 2q^{18} + \cdots \in S_{22}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 5q^{2} + 12q^{3} + 12q^{4} + 11q^{5} + 8q^{6} + 2q^{7} + 8q^{8} + 11q^{9} + 3q^{10} + q^{12} + 4q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(10)\omega^{8}&0\\0&u(4)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1024q^{2} + 59316q^{3} + 1048576q^{4} + 4975350q^{5} + 60739584q^{6} + 1427425832q^{7} + 1073741824q^{8} + -6941965347q^{9} + 5094758400q^{10} + -631762692*13^2q^{11} + 62197334016q^{12} + -11550043498*13q^{13} + 1461684051968q^{14} + 295117860600q^{15} + 1099511627776q^{16} + -11203980739758q^{17} + -7108572515328q^{18} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 10q^{4} + 8q^{5} + 5q^{7} + 8q^{8} + 4q^{9} + 9q^{10} + 2q^{11} + 4q^{14} + 9q^{16} + 4q^{17} + 11q^{18} + \cdots \in S_{24}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 9q^{4} + 12q^{5} + 10q^{7} + q^{8} + 12q^{9} + 10q^{10} + 12q^{11} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{10}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2048q^{2} + -38916*13q^{3} + 4194304q^{4} + -90135570q^{5} + 79699968*13q^{6} + 6872255096q^{7} + -8589934592q^{8} + 161799725637q^{9} + 184597647360q^{10} + -965328798588q^{11} + -163225534464*13q^{12} + 41719999934*13q^{13} + -14074378436608q^{14} + 3507715842120*13q^{15} + 17592186044416q^{16} + 82083537265266q^{17} + -331365838104576q^{18} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 2q^{2} + 12q^{3} + 4q^{4} + 4q^{5} + 11q^{6} + 4q^{7} + 8q^{8} + 8q^{10} + 2q^{11} + 9q^{12} + 8q^{14} + 9q^{15} + 3q^{16} + 7q^{17} + \cdots \in S_{16}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 5q^{2} + 12q^{3} + 12q^{4} + 11q^{5} + 8q^{6} + 2q^{7} + 8q^{8} + 11q^{9} + 3q^{10} + q^{12} + 4q^{13} + 10q^{14} + 2q^{15} + q^{16} + 3q^{17} + 3q^{18} + \cdots \in S_{20}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 9q^{4} + 12q^{5} + 10q^{7} + q^{8} + 12q^{9} + 10q^{10} + 12q^{11} + 7q^{13} + 4q^{14} + 3q^{16} + q^{17} + 10q^{18} + \cdots \in S_{22}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 12q^{2} + q^{3} + q^{4} + 10q^{5} + 12q^{6} + 12q^{7} + 12q^{8} + 11q^{9} + 3q^{10} + 6q^{11} + q^{12} + q^{13} + q^{14} + 10q^{15} + q^{16} + 10q^{17} + 2q^{18} + \cdots \in S_{26}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 10q^{3} + q^{4} + 12q^{5} + 10q^{6} + q^{7} + q^{8} + 6q^{9} + 12q^{10} + 11q^{11} + 10q^{12} + 12q^{13} + q^{14} + 3q^{15} + q^{16} + 10q^{17} + 6q^{18} + \cdots \in S_{26}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 4q^{3} + q^{4} + 6q^{5} + 4q^{6} + 8q^{7} + q^{8} + 6q^{10} + 12q^{11} + 4q^{12} + 8q^{14} + 11q^{15} + q^{16} + 5q^{17} + \cdots \in S_{26}(\Gamma_0(2);\overline{\mathbf{F}}_{13}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 3, \quad \ell = 13}$ \quad (20 forms)}

\begin{enumerate}
\item  Consider
$f = q + 10q^{3} + 5q^{4} + q^{5} + 2q^{7} + 9q^{9} + 3q^{11} + 11q^{12} + 10q^{15} + 12q^{16} + 9q^{19} + 5q^{20} + 7q^{21} + \cdots \in S_{16}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 4q^{3} + 6q^{4} + 12q^{5} + 8q^{7} + 3q^{9} + 4q^{11} + 11q^{12} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(7)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -18*13q^{2} + -2187q^{3} + 21988q^{4} + 280710q^{5} + 39366*13q^{6} + -1373344q^{7} + 194040*13q^{8} + 4782969q^{9} + -5052780*13q^{10} + 34031052q^{11} + -48087756q^{12} + 29540174*13q^{13} + 24720192*13q^{14} + -613912770q^{15} + -1310772464q^{16} + 96862122*13q^{17} + -86093442*13q^{18} + -2499071020q^{19} + 6172251480q^{20} + 3003503328q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 4q^{3} + 10q^{4} + 12q^{5} + 10q^{6} + 6q^{7} + 10q^{8} + 3q^{9} + 4q^{10} + 8q^{11} + q^{12} + 2q^{14} + 9q^{15} + 4q^{16} + 5q^{17} + q^{18} + 8q^{19} + 3q^{20} + 11q^{21} + \cdots \in S_{18}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 10q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + 5q^{7} + 10q^{8} + 9q^{9} + 10q^{10} + 7q^{11} + q^{12} + 6q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(11)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 204q^{2} + -6561q^{3} + -89456q^{4} + -163554q^{5} + -1338444q^{6} + -20846560q^{7} + -44987712q^{8} + 43046721q^{9} + -33365016q^{10} + 817372356q^{11} + 586920816q^{12} + 23045366*13q^{13} + -4252698240q^{14} + 1073077794q^{15} + 2547683584q^{16} + -44775606078q^{17} + 8781531084q^{18} + 78748651964q^{19} + 14630886624q^{20} + 136774280160q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 9q^{3} + 11q^{4} + 4q^{5} + 5q^{6} + 5q^{7} + 10q^{8} + 3q^{9} + 8q^{10} + 8q^{11} + 8q^{12} + 10q^{14} + 10q^{15} + 6q^{16} + 5q^{17} + 6q^{18} + 2q^{19} + 5q^{20} + 6q^{21} + \cdots \in S_{18}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 3q^{3} + 7q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + 10q^{8} + 9q^{9} + 7q^{10} + 7q^{11} + 8q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 6533394632131479889766 + O(13^20)q^{2} + 6561q^{3} + 3823801410415968917440 + O(13^20)q^{4} + -1542937497041889790240 + O(13^20)q^{5} + 9408869058436822258471 + O(13^20)q^{6} + 6128449026840311645739 + O(13^20)q^{7} + 8675214794001793095263 + O(13^20)q^{8} + 43046721q^{9} + 6945025549243229718486 + O(13^20)q^{10} + 56162924892324301556032 + O(13^21)q^{11} + 1408870896516808106520 + O(13^20)q^{12} + -52642131997196450296*13 + O(13^20)q^{13} + -5111726847205962718946 + O(13^20)q^{14} + 6432773919627187116293 + O(13^20)q^{15} + 6447501208163510051225 + O(13^20)q^{16} + 7815365607506701953090 + O(13^20)q^{17} + 3467551591154260602583 + O(13^20)q^{18} + -3038300688986688448893 + O(13^20)q^{19} + 8112039492467298620510 + O(13^20)q^{20} + -5749282548486904809337 + O(13^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + q^{3} + 3q^{4} + 4q^{5} + q^{6} + 6q^{7} + 5q^{8} + q^{9} + 4q^{10} + 10q^{11} + 3q^{12} + 6q^{14} + 4q^{15} + 11q^{16} + 5q^{17} + q^{18} + 5q^{19} + 12q^{20} + 6q^{21} + \cdots \in S_{20}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + 11q^{6} + 12q^{7} + 12q^{8} + 3q^{9} + 3q^{10} + 8q^{11} + 10q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{6}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1104q^{2} + 19683q^{3} + 694528q^{4} + 3516270q^{5} + -21730032q^{6} + -195590584q^{7} + -187944960q^{8} + 387420489q^{9} + -3881962080q^{10} + -2746857948q^{11} + 13670394624q^{12} + -3415418866*13q^{13} + 215932004736q^{14} + 69210742410q^{15} + -156641460224q^{16} + -785982517614q^{17} + -427712219856q^{18} + 315410465180q^{19} + 2442147970560q^{20} + -3849809464872q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 12q^{3} + 12q^{4} + 6q^{6} + 12q^{7} + 7q^{8} + q^{9} + 12q^{11} + q^{12} + 6q^{14} + 8q^{16} + 7q^{17} + 7q^{18} + q^{19} + q^{21} + \cdots \in S_{20}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 6q^{2} + 12q^{3} + 12q^{4} + 7q^{6} + q^{7} + 6q^{8} + q^{9} + q^{11} + q^{12} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(11)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -6747558085137094074388 + O(13^20)q^{2} + -19683q^{3} + -42559723370682578696689 + O(13^21)q^{4} + 7347283092541032338224*13 + O(13^21)q^{5} + 5498930886396187837616 + O(13^20)q^{6} + -803958468900441323964 + O(13^20)q^{7} + 1557510324270151944999 + O(13^20)q^{8} + 387420489q^{9} + -8455029991767779053689*13 + O(13^21)q^{10} + -3326006659352169235353 + O(13^20)q^{11} + -92782982925085173734896 + O(13^21)q^{12} + -324133456109013333717*13 + O(13^20)q^{13} + -79050543617869195250119 + O(13^21)q^{14} + -7803747417136583426183*13 + O(13^21)q^{15} + 584967883585102597254605 + O(13^22)q^{16} + 118829380733358305297788 + O(13^21)q^{17} + -2187938990012403824033 + O(13^20)q^{18} + 2732136477488455389 + O(13^20)q^{19} + -8166293125988553999268*13 + O(13^21)q^{20} + -6820281108319352937821 + O(13^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 12q^{2} + 10q^{3} + 9q^{4} + 3q^{6} + 7q^{7} + 9q^{8} + 9q^{9} + 11q^{11} + 12q^{12} + 6q^{14} + 11q^{16} + 2q^{17} + 4q^{18} + 7q^{19} + 5q^{21} + \cdots \in S_{22}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 6q^{2} + 12q^{3} + 12q^{4} + 7q^{6} + q^{7} + 6q^{8} + q^{9} + q^{11} + q^{12} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{8}&0\\0&u(11)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1728q^{2} + -59049q^{3} + 888832q^{4} + -3193290*13q^{5} + -102036672q^{6} + 538429808q^{7} + -2087976960q^{8} + 3486784401q^{9} + -5518005120*13q^{10} + -64113040188q^{11} + -52484640768q^{12} + -10075392922*13q^{13} + 930406708224q^{14} + 188560581210*13q^{15} + -5472039993344q^{16} + 8242029723618q^{17} + 6025163444928q^{18} + 13492101753020q^{19} + -2838298337280*13q^{20} + -31793741732592q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 11q^{2} + 3q^{3} + 12q^{4} + 6q^{5} + 7q^{6} + 10q^{7} + 12q^{8} + 9q^{9} + q^{10} + 7q^{11} + 10q^{12} + 6q^{14} + 5q^{15} + 7q^{16} + 7q^{17} + 8q^{18} + 9q^{19} + 7q^{20} + 4q^{21} + \cdots \in S_{22}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 7q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + 11q^{6} + 12q^{7} + 12q^{8} + 3q^{9} + 3q^{10} + 8q^{11} + 10q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 5466151690893158575087 + O(13^20)q^{2} + 59049q^{3} + -8496018642269880874010 + O(13^20)q^{4} + 7377492453496132559208 + O(13^20)q^{5} + 9491406749503831154880 + O(13^20)q^{6} + 1792409175580618195143 + O(13^20)q^{7} + -6704706229958762072295 + O(13^20)q^{8} + 3486784401q^{9} + -8825601452609899112765 + O(13^20)q^{10} + -25009939063780530177428 + O(13^21)q^{11} + -7376041865732943386493 + O(13^20)q^{12} + 379367062299107008261*13 + O(13^20)q^{13} + -4550457299176469102143 + O(13^20)q^{14} + 1772238675446752476670 + O(13^20)q^{15} + 1577906784961885665021 + O(13^20)q^{16} + 1045474998125928907635 + O(13^20)q^{17} + 1695430216950414267630 + O(13^20)q^{18} + 4460561882471771995967 + O(13^20)q^{19} + -6275045986857869274852 + O(13^20)q^{20} + 1326146548751730316238 + O(13^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 9q^{3} + 2q^{4} + 7q^{5} + 12q^{6} + q^{7} + 2q^{8} + 3q^{9} + 5q^{10} + 12q^{11} + 5q^{12} + 10q^{14} + 11q^{15} + 6q^{16} + 8q^{17} + 4q^{18} + 10q^{19} + q^{20} + 9q^{21} + \cdots \in S_{24}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 5q^{2} + 3q^{3} + 7q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + 10q^{8} + 9q^{9} + 7q^{10} + 7q^{11} + 8q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{10}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1128q^{2} + 177147q^{3} + -7116224q^{4} + -48863730q^{5} + 199821816q^{6} + -1723688680q^{7} + -17489450496q^{8} + 31381059609q^{9} + -55118287440q^{10} + -1428263180124q^{11} + -1260617732928q^{12} + -632381849602*13q^{13} + -1944320831040q^{14} + -8656063178310q^{15} + 39967113416704q^{16} + -5989210330446q^{17} + 35397835238952q^{18} + 680005481275676q^{19} + 347725248155520q^{20} + -305346278595960q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 4q^{3} + 3q^{4} + 8q^{5} + 11q^{6} + 9q^{7} + 2q^{8} + 3q^{9} + 9q^{10} + 12q^{11} + 12q^{12} + 2q^{14} + 6q^{15} + 4q^{16} + 8q^{17} + 5q^{18} + q^{19} + 11q^{20} + 10q^{21} + \cdots \in S_{24}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 3q^{2} + 10q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + 5q^{7} + 10q^{8} + 9q^{9} + 10q^{10} + 7q^{11} + q^{12} + 6q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(11)\omega^{10}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 7999190284076529919946 + O(13^20)q^{2} + -177147q^{3} + 4601721439607941919351 + O(13^20)q^{4} + -2824075798267790349769 + O(13^20)q^{5} + -3457234417758772232701 + O(13^20)q^{6} + -6855991533239110890125 + O(13^20)q^{7} + 2201679203470269506050 + O(13^20)q^{8} + 31381059609q^{9} + -3047782701909282676616 + O(13^20)q^{10} + 3625538007373198488259 + O(13^20)q^{11} + -1236666265956858780304 + O(13^20)q^{12} + -641482261434181347247*13 + O(13^20)q^{13} + -6145206873657502223558 + O(13^20)q^{14} + 8893989556973462970320 + O(13^20)q^{15} + -5296669873604408233255 + O(13^20)q^{16} + 4361411687787850103520 + O(13^20)q^{17} + 3747757179462790921822 + O(13^20)q^{18} + 9110235842523878111492 + O(13^20)q^{19} + 5085087935422918591751 + O(13^20)q^{20} + 6122104951288716395470 + O(13^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 12q^{3} + 11q^{4} + 8q^{5} + 4q^{7} + q^{9} + 5q^{11} + 2q^{12} + 5q^{15} + 4q^{16} + 8q^{19} + 10q^{20} + 9q^{21} + \cdots \in S_{26}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 4q^{3} + 6q^{4} + 12q^{5} + 8q^{7} + 3q^{9} + 4q^{11} + 11q^{12} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 644367760420693768319*13 + O(13^20)q^{2} + -531441q^{3} + 3632812915992180340275 + O(13^20)q^{4} + 2325514655030757746369 + O(13^20)q^{5} + -314307608669072015845*13 + O(13^20)q^{6} + 1166740301787361687104 + O(13^20)q^{7} + -702100192985395702497*13 + O(13^20)q^{8} + 282429536481q^{9} + -562062436975633545577*13 + O(13^20)q^{10} + 2613135682245105670905 + O(13^20)q^{11} + -6483816534301225486690 + O(13^20)q^{12} + -682759059343875729066*13 + O(13^20)q^{13} + 134779380892237488549*13 + O(13^20)q^{14} + -44467477420782297500 + O(13^20)q^{15} + -9087606788716508980768 + O(13^20)q^{16} + 195076649034533838412*13 + O(13^20)q^{17} + -138679017420541050021*13 + O(13^20)q^{18} + -239624546601476546697 + O(13^20)q^{19} + 2355175398120143292938 + O(13^20)q^{20} + 2315397083680134084562 + O(13^20)q^{21} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 6q^{2} + 3q^{3} + 2q^{4} + 4q^{5} + 5q^{6} + 4q^{7} + 3q^{8} + 9q^{9} + 11q^{10} + 2q^{11} + 6q^{12} + 11q^{14} + 12q^{15} + 2q^{16} + 7q^{17} + 2q^{18} + 3q^{19} + 8q^{20} + 12q^{21} + \cdots \in S_{16}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + 7q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + 11q^{6} + 12q^{7} + 12q^{8} + 3q^{9} + 3q^{10} + 8q^{11} + 10q^{12} + q^{13} + 6q^{14} + 2q^{15} + 8q^{16} + 11q^{17} + 8q^{18} + 3q^{19} + 11q^{20} + 4q^{21} + \cdots \in S_{18}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 12q^{3} + 12q^{4} + 7q^{6} + q^{7} + 6q^{8} + q^{9} + q^{11} + q^{12} + 11q^{13} + 6q^{14} + 8q^{16} + 7q^{17} + 6q^{18} + 12q^{19} + 12q^{21} + \cdots \in S_{20}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 10q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + 5q^{7} + 10q^{8} + 9q^{9} + 10q^{10} + 7q^{11} + q^{12} + 6q^{13} + 2q^{14} + 3q^{15} + 10q^{16} + 2q^{17} + q^{18} + 11q^{19} + 9q^{20} + 11q^{21} + \cdots \in S_{22}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 5q^{2} + 3q^{3} + 7q^{4} + 4q^{5} + 2q^{6} + 2q^{7} + 10q^{8} + 9q^{9} + 7q^{10} + 7q^{11} + 8q^{12} + q^{13} + 10q^{14} + 12q^{15} + 2q^{16} + 2q^{17} + 6q^{18} + 6q^{19} + 2q^{20} + 6q^{21} + \cdots \in S_{22}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{3} + 6q^{4} + 12q^{5} + 8q^{7} + 3q^{9} + 4q^{11} + 11q^{12} + 7q^{13} + 9q^{15} + 10q^{16} + 10q^{19} + 7q^{20} + 6q^{21} + \cdots \in S_{24}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + q^{2} + 12q^{3} + 12q^{4} + 2q^{5} + 12q^{6} + 9q^{7} + 10q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + q^{13} + 9q^{14} + 11q^{15} + 12q^{16} + 2q^{17} + q^{18} + 11q^{20} + 4q^{21} + \cdots \in S_{26}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^102q^{2} + q^{3} + z^89q^{4} + z^21q^{5} + z^102q^{6} + z^105q^{7} + z^167q^{8} + q^{9} + z^123q^{10} + 11q^{11} + z^89q^{12} + 12q^{13} + z^39q^{14} + z^21q^{15} + 3q^{16} + z^163q^{17} + z^102q^{18} + z^21q^{19} + z^110q^{20} + z^105q^{21} + \cdots \in S_{26}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^150q^{2} + q^{3} + z^149q^{4} + z^105q^{5} + z^150q^{6} + z^21q^{7} + z^155q^{8} + q^{9} + z^87q^{10} + 11q^{11} + z^149q^{12} + 12q^{13} + z^3q^{14} + z^105q^{15} + 3q^{16} + z^103q^{17} + z^150q^{18} + z^105q^{19} + z^86q^{20} + z^21q^{21} + \cdots \in S_{26}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + q^{3} + 7q^{4} + 6q^{5} + 3q^{6} + 8q^{7} + 2q^{8} + q^{9} + 5q^{10} + 12q^{11} + 7q^{12} + 11q^{14} + 6q^{15} + 5q^{16} + 5q^{17} + 3q^{18} + 7q^{19} + 3q^{20} + 8q^{21} + \cdots \in S_{26}(\Gamma_0(3);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 4, \quad \ell = 13}$ \quad (10 forms)}

\begin{enumerate}
\item  Consider
$f = q + 10q^{3} + 2q^{7} + 8q^{9} + 11q^{11} + q^{17} + 7q^{19} + 7q^{21} + 9q^{23} + 5q^{25} + \cdots \in S_{16}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 4q^{3} + 8q^{7} + 7q^{9} + 6q^{11} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(11)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -276q^{3} + -10170*13q^{5} + -3585736q^{7} + -14272731q^{9} + 47801700q^{11} + 19060382*13q^{13} + 2806920*13q^{15} + -2127682062q^{17} + -1074862756q^{19} + 989663136q^{21} + 24982896168q^{23} + -13038094025q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{3} + 10q^{5} + q^{7} + 3q^{9} + 2q^{15} + 9q^{17} + 6q^{19} + 8q^{21} + 11q^{23} + 4q^{25} + \cdots \in S_{18}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{3} + 10q^{5} + 3q^{7} + 9q^{9} + 6q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(11)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1286516409746090632052 + O(13^20)q^{3} + -8304663201097663680088 + O(13^20)q^{5} + 3992196393945872308288 + O(13^20)q^{7} + -740906904454532522821 + O(13^20)q^{9} + 212801406137125418280*13 + O(13^20)q^{11} + 648160538609243686552*13 + O(13^20)q^{13} + -43903840148628846031332 + O(13^21)q^{15} + -105754244764442019312571 + O(13^21)q^{17} + 617212345240815903988 + O(13^20)q^{19} + -8612919168897456738590 + O(13^20)q^{21} + 3874938670922714935421 + O(13^20)q^{23} + -2785300275674232115146 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{3} + 10q^{5} + 8q^{7} + 6q^{9} + 12q^{11} + 4q^{15} + 10q^{17} + 11q^{19} + 11q^{21} + q^{23} + q^{25} + \cdots \in S_{20}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{3} + 2q^{5} + 3q^{7} + 5q^{9} + 7q^{11} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{6}&0\\0&u(11)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -36q^{3} + -196290q^{5} + -35905576q^{7} + -1162260171q^{9} + -12016099980q^{11} + -3502281298*13q^{13} + 7066440q^{15} + 496563248178q^{17} + 1410273986444q^{19} + 1292600736q^{21} + -7039745388792q^{23} + -19034956564025q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{3} + 2q^{5} + 9q^{7} + 2q^{9} + 11q^{11} + 5q^{15} + q^{17} + 12q^{19} + 3q^{21} + 10q^{23} + 12q^{25} + \cdots \in S_{22}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + q^{3} + 2q^{5} + 3q^{7} + 5q^{9} + 7q^{11} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(11)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 8462545186296674284210 + O(13^20)q^{3} + -3092485509631194802669 + O(13^20)q^{5} + 1469087420350555161841 + O(13^20)q^{7} + 4384816297698094731553 + O(13^20)q^{9} + -6013201347648037186074 + O(13^20)q^{11} + 575156692355116302526*13 + O(13^20)q^{13} + 19023796711690116237481 + O(13^22)q^{15} + 20837583580866031349800 + O(13^21)q^{17} + -557610737800364088400 + O(13^20)q^{19} + -3279839831451634945920 + O(13^20)q^{21} + -5482614947438987743349 + O(13^20)q^{23} + -4174005067286584637056 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{3} + 11q^{5} + 8q^{7} + 3q^{9} + 10q^{15} + 4q^{17} + 4q^{19} + 12q^{21} + 2q^{23} + 9q^{25} + \cdots \in S_{24}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 7q^{3} + 10q^{5} + 3q^{7} + 9q^{9} + 6q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(11)\omega^{10}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1331450926767302681374 + O(13^20)q^{3} + 3205122991126930639868 + O(13^20)q^{5} + 86325831963985043709261 + O(13^21)q^{7} + 5714777634412893382443 + O(13^20)q^{9} + 336883763590512412823*13 + O(13^20)q^{11} + 654417486999956679088*13 + O(13^20)q^{13} + -8641147596384797105290 + O(13^20)q^{15} + -9039199475879134148739 + O(13^20)q^{17} + 8769950732729624650528 + O(13^20)q^{19} + 7450228016826599530785 + O(13^20)q^{21} + 2244806087364882828971 + O(13^20)q^{23} + 3087593104344696290768 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 12q^{3} + 4q^{7} + 11q^{9} + q^{11} + 10q^{17} + 12q^{19} + 9q^{21} + 10q^{23} + 8q^{25} + \cdots \in S_{26}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 4q^{3} + 8q^{7} + 7q^{9} + 6q^{11} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{0}&0\\0&u(11)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -8650085295057409311837 + O(13^20)q^{3} + -125783729197557520988*13 + O(13^20)q^{5} + -3764039380953877673150 + O(13^20)q^{7} + 217945006786812127431 + O(13^20)q^{9} + 7481104591237751251454 + O(13^20)q^{11} + -470320238802562194668*13 + O(13^20)q^{13} + -2059659876732524918447*13 + O(13^21)q^{15} + 4886493467816888617011 + O(13^20)q^{17} + -8424716420627594589815 + O(13^20)q^{19} + 193938970300302612693 + O(13^20)q^{21} + -4991984407306657099420 + O(13^20)q^{23} + 3911837401795252551279 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + q^{3} + 2q^{5} + 3q^{7} + 5q^{9} + 7q^{11} + 11q^{13} + 2q^{15} + 9q^{17} + 4q^{19} + 3q^{21} + 4q^{23} + 12q^{25} + \cdots \in S_{18}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{3} + 10q^{5} + 3q^{7} + 9q^{9} + 6q^{13} + 5q^{15} + q^{17} + 5q^{19} + 8q^{21} + 8q^{23} + 4q^{25} + \cdots \in S_{22}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{3} + 8q^{7} + 7q^{9} + 6q^{11} + 11q^{13} + 9q^{17} + 2q^{19} + 6q^{21} + q^{23} + 5q^{25} + \cdots \in S_{24}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 2q^{5} + 11q^{7} + 10q^{9} + 11q^{11} + 12q^{13} + 6q^{17} + 7q^{19} + 8q^{23} + 12q^{25} + \cdots \in S_{26}(\Gamma_0(4);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 5, \quad \ell = 13}$ \quad (37 forms)}

\begin{enumerate}
\item  Consider
$f = q + 6q^{2} + 9q^{3} + 2q^{4} + 8q^{5} + 2q^{6} + 5q^{7} + 3q^{8} + 2q^{9} + 9q^{10} + 7q^{11} + 5q^{12} + 4q^{14} + 7q^{15} + 2q^{16} + 5q^{17} + 12q^{18} + 9q^{19} + 3q^{20} + 6q^{21} + 3q^{22} + 3q^{23} + q^{24} + 12q^{25} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 8q^{2} + q^{3} + 5q^{4} + 5q^{5} + 8q^{6} + 7q^{7} + 10q^{8} + 5q^{9} + q^{10} + 5q^{11} + 5q^{12} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(7)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -7317555275418938706865 + O(13^20)q^{2} + 4369853224042922026251 + O(13^20)q^{3} + 6851446169055865882559 + O(13^20)q^{4} + 78125q^{5} + 7241221139647002945608 + O(13^20)q^{6} + -8884759449514947803865 + O(13^20)q^{7} + -5481959072079439705961 + O(13^20)q^{8} + 1559099882364535164617 + O(13^20)q^{9} + 4309420084741444744756 + O(13^20)q^{10} + 6337551149897323847069 + O(13^20)q^{11} + 4586322503949046117067 + O(13^20)q^{12} + 310809223956986403598*13 + O(13^20)q^{13} + -5851625339545966354513 + O(13^20)q^{14} + 8618840169482981677012 + O(13^20)q^{15} + 2470238417902660021437 + O(13^20)q^{16} + 1650161398412153928384 + O(13^20)q^{17} + -9346877808115782954670 + O(13^20)q^{18} + -93774431108904567930458 + O(13^21)q^{19} + -5572762028194118908290 + O(13^20)q^{20} + -4979397372529051051574 + O(13^20)q^{21} + -3151619181712356784465 + O(13^20)q^{22} + -9204305260627694932452 + O(13^20)q^{23} + 3018445152997214475432 + O(13^20)q^{24} + 6103515625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^99q^{2} + z^79q^{3} + z^64q^{4} + 5q^{5} + z^10q^{6} + z^67q^{7} + z^48q^{8} + z^127q^{9} + z^57q^{10} + z^119q^{11} + z^143q^{12} + z^166q^{14} + z^37q^{15} + z^86q^{16} + z^7q^{17} + z^58q^{18} + z^22q^{20} + z^146q^{21} + z^50q^{22} + z^54q^{23} + z^127q^{24} + 12q^{25} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^71q^{2} + z^135q^{3} + z^8q^{4} + 8q^{5} + z^38q^{6} + z^95q^{7} + z^132q^{8} + z^71q^{9} + z^113q^{10} + z^91q^{11} + z^143q^{12} + z^123q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^45)\omega^{1}&0\\0&u(z^123)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (5220066930185672060144 + O(13^20))*a - 3930958870456881499590 + O(13^20)q^{3} + -(5872575296458881067501 + O(13^20))*a + 4422328729263991694938 + O(13^20)q^{4} + -78125q^{5} + -(2337065719276137568291 + O(13^20))*a + 9001701865544814262000 + O(13^20)q^{6} + -(8187241859314354706443 + O(13^20))*a - 8490095368981084080962 + O(13^20)q^{7} + -(4485337410954724837403 + O(13^20))*a - 1315648857824827119721 + O(13^20)q^{8} + -(2266125169566136746825 + O(13^20))*a + 455518494574115769770*13 + O(13^20)q^{9} + -78125*aq^{10} + (8617762588546044810657 + O(13^20))*a - 7431882862003163593015 + O(13^20)q^{11} + -(2152228184105116914181 + O(13^20))*a + 3094366649193076852094 + O(13^20)q^{12} + (165960082856164194028*13 + O(13^21))*a + 4306357358057521888581*13 + O(13^21)q^{13} + -(8658264380083863399096 + O(13^20))*a - 4249744541549648999608 + O(13^20)q^{14} + -(9216239363435340958142 + O(13^20))*a + 4952116145029023883391 + O(13^20)q^{15} + (4808953201033301535716 + O(13^20))*a + 2442993228561431063000 + O(13^20)q^{16} + -(5954622859256493986977 + O(13^20))*a - 1868043271838626439554 + O(13^20)q^{17} + (5459188477617129926663 + O(13^20))*a + 732326720394207323054 + O(13^20)q^{18} + -(42377726526926250859*13 + O(13^20))*a + 491889676940628709447*13 + O(13^20)q^{19} + -(3885453547295853579316 + O(13^20))*a - 3195510191298169067871 + O(13^20)q^{20} + (2860399369675169900761 + O(13^20))*a + 34669688693115936688 + O(13^20)q^{21} + -(12595736001393929554795 + O(13^21))*a - 112823376207622871642925 + O(13^21)q^{22} + (1588112852478991517499 + O(13^20))*a - 4539486969322660788060 + O(13^20)q^{23} + (9045159991055509217667 + O(13^20))*a - 685631717776809004374*13 + O(13^20)q^{24} + 6103515625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 5872575296458881067501 + O(13^20)x + -4422328729263991727706 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^111q^{2} + z^19q^{3} + z^160q^{4} + 5q^{5} + z^130q^{6} + z^31q^{7} + z^120q^{8} + z^139q^{9} + z^69q^{10} + z^35q^{11} + z^11q^{12} + z^142q^{14} + z^145q^{15} + z^110q^{16} + z^91q^{17} + z^82q^{18} + z^118q^{20} + z^50q^{21} + z^146q^{22} + z^30q^{23} + z^139q^{24} + 12q^{25} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^83q^{2} + z^75q^{3} + z^104q^{4} + 8q^{5} + z^158q^{6} + z^59q^{7} + z^36q^{8} + z^83q^{9} + z^125q^{10} + z^7q^{11} + z^11q^{12} + z^87q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^81)\omega^{1}&0\\0&u(z^87)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (5220066930185672060144 + O(13^20))*a - 3930958870456881499590 + O(13^20)q^{3} + -(5872575296458881067501 + O(13^20))*a + 4422328729263991694938 + O(13^20)q^{4} + -78125q^{5} + -(2337065719276137568291 + O(13^20))*a + 9001701865544814262000 + O(13^20)q^{6} + -(8187241859314354706443 + O(13^20))*a - 8490095368981084080962 + O(13^20)q^{7} + -(4485337410954724837403 + O(13^20))*a - 1315648857824827119721 + O(13^20)q^{8} + -(2266125169566136746825 + O(13^20))*a + 455518494574115769770*13 + O(13^20)q^{9} + -78125*aq^{10} + (8617762588546044810657 + O(13^20))*a - 7431882862003163593015 + O(13^20)q^{11} + -(2152228184105116914181 + O(13^20))*a + 3094366649193076852094 + O(13^20)q^{12} + (165960082856164194028*13 + O(13^21))*a + 4306357358057521888581*13 + O(13^21)q^{13} + -(8658264380083863399096 + O(13^20))*a - 4249744541549648999608 + O(13^20)q^{14} + -(9216239363435340958142 + O(13^20))*a + 4952116145029023883391 + O(13^20)q^{15} + (4808953201033301535716 + O(13^20))*a + 2442993228561431063000 + O(13^20)q^{16} + -(5954622859256493986977 + O(13^20))*a - 1868043271838626439554 + O(13^20)q^{17} + (5459188477617129926663 + O(13^20))*a + 732326720394207323054 + O(13^20)q^{18} + -(42377726526926250859*13 + O(13^20))*a + 491889676940628709447*13 + O(13^20)q^{19} + -(3885453547295853579316 + O(13^20))*a - 3195510191298169067871 + O(13^20)q^{20} + (2860399369675169900761 + O(13^20))*a + 34669688693115936688 + O(13^20)q^{21} + -(12595736001393929554795 + O(13^21))*a - 112823376207622871642925 + O(13^21)q^{22} + (1588112852478991517499 + O(13^20))*a - 4539486969322660788060 + O(13^20)q^{23} + (9045159991055509217667 + O(13^20))*a - 685631717776809004374*13 + O(13^20)q^{24} + 6103515625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 5872575296458881067501 + O(13^20)x + -4422328729263991727706 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 9q^{3} + 11q^{4} + 12q^{5} + 5q^{6} + 10q^{7} + 10q^{8} + 7q^{9} + 11q^{10} + 8q^{11} + 8q^{12} + 7q^{14} + 4q^{15} + 6q^{16} + 12q^{17} + q^{18} + 7q^{19} + 2q^{20} + 12q^{21} + 3q^{22} + 12q^{23} + 12q^{24} + q^{25} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{12} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 9q^{3} + 11q^{4} + 12q^{5} + 5q^{6} + 10q^{7} + 10q^{8} + 7q^{9} + 11q^{10} + 8q^{11} + 8q^{12} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{5}$. The form
$f_0 = q + aq^{2} + -4*13*a + 12190q^{3} + 680*a - 196128q^{4} + -390625q^{5} + -23170*a + 260224*13q^{6} + -3668*13*a + 4802210q^{7} + 800*13^2*a - 44238080q^{8} + 43920*13*a - 156455487q^{9} + -390625*aq^{10} + -895000*a - 222377728q^{11} + -5556944*a - 90420160q^{12} + 19322608*a - 7388399550q^{13} + -27622910*a + 238625408*13q^{14} + 1562500*13*a - 4761718750q^{15} + -41431040*a + 16911318016q^{16} + 49420496*a + 5839310330q^{17} + 231797313*a - 2857259520*13q^{18} + -356859680*a + 124815536700q^{19} + -265625000*a + 76612500000q^{20} + 65778720*13*a - 102771835908q^{21} + -830977728*a + 58225120000q^{22} + 6400036*13*a + 7811623910q^{23} + -64015680*13*a - 81892492800q^{24} + 152587890625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -680x + 65056$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + 5q^{2} + 6q^{3} + 6q^{4} + q^{5} + 4q^{6} + 3q^{7} + q^{9} + 5q^{10} + q^{11} + 10q^{12} + 2q^{14} + 6q^{15} + 3q^{16} + 5q^{18} + 2q^{19} + 6q^{20} + 5q^{21} + 5q^{22} + q^{25} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 9q^{2} + 2q^{3} + 8q^{4} + 8q^{5} + 5q^{6} + 6q^{7} + 3q^{9} + 7q^{10} + 6q^{11} + 3q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{4}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -3628868016030687910891 + O(13^20)q^{2} + -5504779284713785909918 + O(13^20)q^{3} + 1023525606569183131550 + O(13^20)q^{4} + 390625q^{5} + 7488231323307237180773 + O(13^20)q^{6} + -2966270903751734366148 + O(13^20)q^{7} + 519765099132085556010*13 + O(13^20)q^{8} + -8600398090149394577073 + O(13^20)q^{9} + -3335684953277449946688 + O(13^20)q^{10} + -4963304452297732829508 + O(13^20)q^{11} + -3391291486846302714223 + O(13^20)q^{12} + 8916898538001502833158*13 + O(13^21)q^{13} + -2023094226239506021964 + O(13^20)q^{14} + -6786746209449358018406 + O(13^20)q^{15} + 6396786447117712380062 + O(13^20)q^{16} + 25588113189823371428*13^2 + O(13^20)q^{17} + -8267642963816146769750 + O(13^20)q^{18} + 195426920231191621273 + O(13^20)q^{19} + 7267133919782967662113 + O(13^20)q^{20} + 6019343683582895058567 + O(13^20)q^{21} + 3408641427768813938915 + O(13^20)q^{22} + -693084030206458736165*13 + O(13^20)q^{23} + -370345835700916827148*13 + O(13^20)q^{24} + 152587890625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^142q^{2} + z^91q^{3} + z^26q^{4} + q^{5} + z^65q^{6} + z^50q^{7} + z^147q^{8} + 6q^{9} + z^142q^{10} + z^34q^{11} + z^117q^{12} + z^24q^{14} + z^91q^{15} + z^114q^{16} + z^31q^{17} + z^44q^{18} + z^50q^{19} + z^26q^{20} + z^141q^{21} + z^8q^{22} + z^25q^{23} + 6q^{24} + q^{25} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^86q^{2} + z^35q^{3} + z^82q^{4} + q^{5} + z^121q^{6} + z^106q^{7} + z^147q^{8} + 5q^{9} + z^86q^{10} + z^146q^{11} + z^117q^{12} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(11)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(8897670551435285067606 + O(13^20))*a + 4481529849824804981044 + O(13^20)q^{3} + (3628868016030687911009 + O(13^20))*a + 7884215324068030411695 + O(13^20)q^{4} + 390625q^{5} + (1023249434888980944818 + O(13^20))*a - 4565751906732634548755 + O(13^20)q^{6} + (3686433389151871135188 + O(13^20))*a + 8628463238335466319478 + O(13^20)q^{7} + -(8907740930637213657401 + O(13^20))*a - 905816729131560384215 + O(13^20)q^{8} + (456938160840024630069*13 + O(13^20))*a - 1870646085682006081638 + O(13^20)q^{9} + 390625*aq^{10} + (5893308489695306725060 + O(13^20))*a - 2664167686264396340838 + O(13^20)q^{11} + -(7050007251147785820815 + O(13^20))*a - 2338166414383750692459 + O(13^20)q^{12} + -(502213623340343758699*13^2 + O(13^21))*a - 4529083645657792988525*13 + O(13^21)q^{13} + -(5662192334583729814022 + O(13^20))*a - 1735998064830012149610 + O(13^20)q^{14} + -(5779040432747366228069 + O(13^20))*a - 6630607780632985964013 + O(13^20)q^{15} + (7393630982736995746305 + O(13^20))*a + 146228111682366633866*13 + O(13^20)q^{16} + -(2017892231264477554405 + O(13^20))*a + 1326919908701455567470 + O(13^20)q^{17} + -(8533919599049074887651 + O(13^20))*a - 477002432080175539617*13 + O(13^20)q^{18} + -(9010191186037033431899 + O(13^20))*a + 8771420192295613525227 + O(13^20)q^{19} + (3335684953277496040438 + O(13^20))*a - 1773719134050288781476 + O(13^20)q^{20} + -(72521969090526922888 + O(13^20))*a - 4678739695580296151890 + O(13^20)q^{21} + (7627472138563918917862 + O(13^20))*a + 6756498618780231762603 + O(13^20)q^{22} + -(4135906859789091867998 + O(13^20))*a - 169602222772110909996 + O(13^20)q^{23} + (339145189732646616887*13 + O(13^20))*a + 2104964063673080765096 + O(13^20)q^{24} + 152587890625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3628868016030687911009 + O(13^20)x + -7884215324068030542767 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^166q^{2} + z^7q^{3} + z^2q^{4} + q^{5} + z^5q^{6} + z^146q^{7} + z^63q^{8} + 6q^{9} + z^166q^{10} + z^106q^{11} + z^9q^{12} + z^144q^{14} + z^7q^{15} + z^138q^{16} + z^67q^{17} + z^68q^{18} + z^146q^{19} + z^2q^{20} + z^153q^{21} + z^104q^{22} + z^157q^{23} + 6q^{24} + q^{25} + \cdots \in S_{18}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^110q^{2} + z^119q^{3} + z^58q^{4} + q^{5} + z^61q^{6} + z^34q^{7} + z^63q^{8} + 5q^{9} + z^110q^{10} + z^50q^{11} + z^9q^{12} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{1}&0\\0&u(11)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(8897670551435285067606 + O(13^20))*a + 4481529849824804981044 + O(13^20)q^{3} + (3628868016030687911009 + O(13^20))*a + 7884215324068030411695 + O(13^20)q^{4} + 390625q^{5} + (1023249434888980944818 + O(13^20))*a - 4565751906732634548755 + O(13^20)q^{6} + (3686433389151871135188 + O(13^20))*a + 8628463238335466319478 + O(13^20)q^{7} + -(8907740930637213657401 + O(13^20))*a - 905816729131560384215 + O(13^20)q^{8} + (456938160840024630069*13 + O(13^20))*a - 1870646085682006081638 + O(13^20)q^{9} + 390625*aq^{10} + (5893308489695306725060 + O(13^20))*a - 2664167686264396340838 + O(13^20)q^{11} + -(7050007251147785820815 + O(13^20))*a - 2338166414383750692459 + O(13^20)q^{12} + -(502213623340343758699*13^2 + O(13^21))*a - 4529083645657792988525*13 + O(13^21)q^{13} + -(5662192334583729814022 + O(13^20))*a - 1735998064830012149610 + O(13^20)q^{14} + -(5779040432747366228069 + O(13^20))*a - 6630607780632985964013 + O(13^20)q^{15} + (7393630982736995746305 + O(13^20))*a + 146228111682366633866*13 + O(13^20)q^{16} + -(2017892231264477554405 + O(13^20))*a + 1326919908701455567470 + O(13^20)q^{17} + -(8533919599049074887651 + O(13^20))*a - 477002432080175539617*13 + O(13^20)q^{18} + -(9010191186037033431899 + O(13^20))*a + 8771420192295613525227 + O(13^20)q^{19} + (3335684953277496040438 + O(13^20))*a - 1773719134050288781476 + O(13^20)q^{20} + -(72521969090526922888 + O(13^20))*a - 4678739695580296151890 + O(13^20)q^{21} + (7627472138563918917862 + O(13^20))*a + 6756498618780231762603 + O(13^20)q^{22} + -(4135906859789091867998 + O(13^20))*a - 169602222772110909996 + O(13^20)q^{23} + (339145189732646616887*13 + O(13^20))*a + 2104964063673080765096 + O(13^20)q^{24} + 152587890625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3628868016030687911009 + O(13^20)x + -7884215324068030542767 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 4q^{3} + 3q^{4} + 5q^{5} + 4q^{6} + 6q^{7} + 5q^{8} + 5q^{10} + 10q^{11} + 12q^{12} + 6q^{14} + 7q^{15} + 11q^{16} + 5q^{17} + 5q^{19} + 2q^{20} + 11q^{21} + 10q^{22} + 11q^{23} + 7q^{24} + 12q^{25} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 12q^{2} + 4q^{3} + 3q^{4} + 8q^{5} + 9q^{6} + 7q^{7} + 8q^{8} + 5q^{10} + 3q^{11} + 12q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -3997202943981199501626 + O(13^20)q^{2} + -3815922344508578977188 + O(13^20)q^{3} + 830545956429987925520 + O(13^20)q^{4} + 1953125q^{5} + 3703085128421191167555 + O(13^20)q^{6} + 7213553689463891027446 + O(13^20)q^{7} + 9019033054024800675107 + O(13^20)q^{8} + 704528137250708010859*13 + O(13^20)q^{9} + -6935843771555947657261 + O(13^20)q^{10} + 1332029656160716152988 + O(13^20)q^{11} + -1906772986591111193270 + O(13^20)q^{12} + 59663741439041826656*13 + O(13^20)q^{13} + -1839997685510947869628 + O(13^20)q^{14} + -5740124838887699551141 + O(13^20)q^{15} + -8982213938502413755343 + O(13^20)q^{16} + -1692752069010160812246 + O(13^20)q^{17} + -665652996891829027619*13 + O(13^20)q^{18} + -4394520728234124610026 + O(13^20)q^{19} + -8611852630469067609355 + O(13^20)q^{20} + 2075544623961906833708 + O(13^20)q^{21} + 2870473235565059563241 + O(13^20)q^{22} + 5531446455694287795067 + O(13^20)q^{23} + -8582109836927375781623 + O(13^20)q^{24} + 3814697265625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + q^{3} + 5q^{4} + 8q^{5} + 4q^{6} + 5q^{7} + 2q^{8} + 11q^{9} + 6q^{10} + 10q^{11} + 5q^{12} + 7q^{14} + 8q^{15} + 5q^{16} + 9q^{17} + 5q^{18} + 3q^{19} + q^{20} + 5q^{21} + q^{22} + 3q^{23} + 2q^{24} + 12q^{25} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 9q^{3} + 11q^{4} + 12q^{5} + 5q^{6} + 10q^{7} + 10q^{8} + 7q^{9} + 11q^{10} + 8q^{11} + 8q^{12} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{19}$. The form
$f_0 = q + 590916624805747362321 + O(13^20)q^{2} + -5620525212896564496018 + O(13^20)q^{3} + 2275749366404628727722 + O(13^20)q^{4} + -1953125q^{5} + 1082101236754686018342 + O(13^20)q^{6} + -5225716572032899906010 + O(13^20)q^{7} + -3225429355888714982356 + O(13^20)q^{8} + -1557602863956814317866 + O(13^20)q^{9} + -592702764128713695997 + O(13^20)q^{10} + 2920048328496090125708 + O(13^20)q^{11} + -1926296774862491398284 + O(13^20)q^{12} + -54726844501180126467*13^2 + O(13^20)q^{13} + 4618759515079135482245 + O(13^20)q^{14} + -1854316720198156760967 + O(13^20)q^{15} + 4641252644652018795795 + O(13^20)q^{16} + -3126970046856470374707 + O(13^20)q^{17} + -1236823702288867187776 + O(13^20)q^{18} + -5993116319547924990719 + O(13^20)q^{19} + 931518756246516430227 + O(13^20)q^{20} + -1909955612149109668879 + O(13^20)q^{21} + -3104831619741225211694 + O(13^20)q^{22} + -3108168701517293773343 + O(13^20)q^{23} + 8750837984708492009768 + O(13^20)q^{24} + 3814697265625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + z^31q^{2} + z^155q^{3} + z^10q^{4} + 8q^{5} + z^18q^{6} + z^142q^{7} + z^110q^{8} + z^69q^{9} + z^73q^{10} + z^132q^{11} + z^165q^{12} + z^5q^{14} + z^29q^{15} + z^48q^{16} + z^114q^{17} + z^100q^{18} + z^71q^{19} + z^52q^{20} + z^129q^{21} + z^163q^{22} + z^106q^{23} + z^97q^{24} + 12q^{25} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^115q^{2} + z^155q^{3} + z^10q^{4} + 5q^{5} + z^102q^{6} + z^58q^{7} + z^26q^{8} + z^69q^{9} + z^73q^{10} + z^48q^{11} + z^165q^{12} + z^80q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^88)\omega^{1}&0\\0&u(z^80)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (8849054544638271778316 + O(13^20))*a + 7393445151983381691030 + O(13^20)q^{3} + (665468732674315613452 + O(13^20))*a - 8612220786347579753582 + O(13^20)q^{4} + -1953125q^{5} + -(7165144223036931759022 + O(13^20))*a - 1451518083788802409923 + O(13^20)q^{6} + -(8438625943682890625367 + O(13^20))*a - 6612631716691744846493 + O(13^20)q^{7} + -(5270232639785987670248 + O(13^20))*a + 1625854945821294370753 + O(13^20)q^{8} + -(5254859995511055602617 + O(13^20))*a + 7318334516511865494137 + O(13^20)q^{9} + -1953125*aq^{10} + -(1748929593451908841899 + O(13^20))*a + 4607457042881780549097 + O(13^20)q^{11} + -(254691389356904123648 + O(13^20))*a + 5204795227336880142202 + O(13^20)q^{12} + (260234160304804397174*13 + O(13^20))*a - 479634133704909184154*13 + O(13^20)q^{13} + (5182820113049397322540 + O(13^20))*a + 3649983210569739410778 + O(13^20)q^{14} + -(5550982276749383420090 + O(13^20))*a - 8996965163097302070532 + O(13^20)q^{15} + (8061152260853551789803 + O(13^20))*a + 3729859948412920961133 + O(13^20)q^{16} + -(3583351279466826108786 + O(13^20))*a - 2925297003712872186434 + O(13^20)q^{17} + (1686985168227981882046 + O(13^20))*a + 5007917712907574784834 + O(13^20)q^{18} + (1476366789493103655230 + O(13^20))*a - 44355410741935931715 + O(13^20)q^{19} + (5854059575191096162890 + O(13^20))*a + 1430137592666115803879 + O(13^20)q^{20} + -(88619093300289800099464 + O(13^21))*a - 8747386396886377526729*13 + O(13^21)q^{21} + (7659469304783353745214 + O(13^20))*a + 909695618483026268329 + O(13^20)q^{22} + (3884137569396310681265 + O(13^20))*a - 3024443866355099996051 + O(13^20)q^{23} + (8112648615256407171147 + O(13^20))*a - 3222010919335750576232 + O(13^20)q^{24} + 3814697265625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -665468732674315613452 + O(13^20)x + 8612220786347579229294 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^67q^{2} + z^167q^{3} + z^130q^{4} + 8q^{5} + z^66q^{6} + z^166q^{7} + z^86q^{8} + z^57q^{9} + z^109q^{10} + z^36q^{11} + z^129q^{12} + z^65q^{14} + z^41q^{15} + z^120q^{16} + z^138q^{17} + z^124q^{18} + z^83q^{19} + z^4q^{20} + z^165q^{21} + z^103q^{22} + z^34q^{23} + z^85q^{24} + 12q^{25} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^151q^{2} + z^167q^{3} + z^130q^{4} + 5q^{5} + z^150q^{6} + z^82q^{7} + z^2q^{8} + z^57q^{9} + z^109q^{10} + z^120q^{11} + z^129q^{12} + z^32q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^136)\omega^{1}&0\\0&u(z^32)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (8849054544638271778316 + O(13^20))*a + 7393445151983381691030 + O(13^20)q^{3} + (665468732674315613452 + O(13^20))*a - 8612220786347579753582 + O(13^20)q^{4} + -1953125q^{5} + -(7165144223036931759022 + O(13^20))*a - 1451518083788802409923 + O(13^20)q^{6} + -(8438625943682890625367 + O(13^20))*a - 6612631716691744846493 + O(13^20)q^{7} + -(5270232639785987670248 + O(13^20))*a + 1625854945821294370753 + O(13^20)q^{8} + -(5254859995511055602617 + O(13^20))*a + 7318334516511865494137 + O(13^20)q^{9} + -1953125*aq^{10} + -(1748929593451908841899 + O(13^20))*a + 4607457042881780549097 + O(13^20)q^{11} + -(254691389356904123648 + O(13^20))*a + 5204795227336880142202 + O(13^20)q^{12} + (260234160304804397174*13 + O(13^20))*a - 479634133704909184154*13 + O(13^20)q^{13} + (5182820113049397322540 + O(13^20))*a + 3649983210569739410778 + O(13^20)q^{14} + -(5550982276749383420090 + O(13^20))*a - 8996965163097302070532 + O(13^20)q^{15} + (8061152260853551789803 + O(13^20))*a + 3729859948412920961133 + O(13^20)q^{16} + -(3583351279466826108786 + O(13^20))*a - 2925297003712872186434 + O(13^20)q^{17} + (1686985168227981882046 + O(13^20))*a + 5007917712907574784834 + O(13^20)q^{18} + (1476366789493103655230 + O(13^20))*a - 44355410741935931715 + O(13^20)q^{19} + (5854059575191096162890 + O(13^20))*a + 1430137592666115803879 + O(13^20)q^{20} + -(88619093300289800099464 + O(13^21))*a - 8747386396886377526729*13 + O(13^21)q^{21} + (7659469304783353745214 + O(13^20))*a + 909695618483026268329 + O(13^20)q^{22} + (3884137569396310681265 + O(13^20))*a - 3024443866355099996051 + O(13^20)q^{23} + (8112648615256407171147 + O(13^20))*a - 3222010919335750576232 + O(13^20)q^{24} + 3814697265625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -665468732674315613452 + O(13^20)x + 8612220786347579229294 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 11q^{2} + 12q^{3} + 12q^{4} + q^{5} + 2q^{6} + 10q^{7} + 12q^{8} + 11q^{10} + 7q^{11} + q^{12} + 6q^{14} + 12q^{15} + 7q^{16} + 7q^{17} + 9q^{19} + 12q^{20} + 3q^{21} + 12q^{22} + 6q^{23} + q^{24} + q^{25} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 12q^{2} + 4q^{3} + 3q^{4} + 8q^{5} + 9q^{6} + 7q^{7} + 8q^{8} + 5q^{10} + 3q^{11} + 12q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{8}&0\\0&u(1)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1819554810271021127956 + O(13^20)q^{2} + -8328385053544572625493 + O(13^20)q^{3} + -1479175213448935642160 + O(13^20)q^{4} + -9765625q^{5} + 3780268403966995216369 + O(13^20)q^{6} + 3760260595160442877839 + O(13^20)q^{7} + 5583720152748733494615 + O(13^20)q^{8} + -24282554209330779441*13 + O(13^20)q^{9} + -41511135229399897271 + O(13^20)q^{10} + 1862874499978548942347 + O(13^20)q^{11} + 3330858244866728329807 + O(13^20)q^{12} + -517015767100422140247*13 + O(13^20)q^{13} + 6781005475680848174925 + O(13^20)q^{14} + -9225791363198914311783 + O(13^20)q^{15} + 4987446201851936178597 + O(13^20)q^{16} + 606595826770047085082 + O(13^20)q^{17} + 140821471560980825916*13 + O(13^20)q^{18} + -4080218807728687044347 + O(13^20)q^{19} + 5637391162662618151532 + O(13^20)q^{20} + -8147337207873222479494 + O(13^20)q^{21} + -7520349561707154082860 + O(13^20)q^{22} + 3709709954636947570549 + O(13^20)q^{23} + -7074878713167804721233 + O(13^20)q^{24} + 95367431640625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + 3q^{3} + 7q^{4} + 12q^{5} + 2q^{6} + 4q^{7} + 10q^{8} + 8q^{9} + 8q^{10} + 7q^{11} + 8q^{12} + 7q^{14} + 10q^{15} + 2q^{16} + 10q^{17} + q^{18} + 8q^{19} + 6q^{20} + 12q^{21} + 9q^{22} + 4q^{23} + 4q^{24} + q^{25} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 9q^{3} + 11q^{4} + 12q^{5} + 5q^{6} + 10q^{7} + 10q^{8} + 7q^{9} + 11q^{10} + 8q^{11} + 8q^{12} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{117}$. The form
$f_0 = q + aq^{2} + (387707808244872145507*13 + O(13^20))*a + 4495394509652086036363 + O(13^20)q^{3} + (768285134085983183771 + O(13^20))*a - 7674192862009659033624 + O(13^20)q^{4} + 9765625q^{5} + -(956715252372160439792 + O(13^20))*a + 279567864209877498264*13 + O(13^20)q^{6} + -(113222794681017468079*13 + O(13^20))*a + 3383181363578729779629 + O(13^20)q^{7} + -(39536137854109273472*13 + O(13^20))*a + 4940991148828087424211 + O(13^20)q^{8} + (234573022357938683876*13 + O(13^20))*a + 5740774548035120505738 + O(13^20)q^{9} + 9765625*aq^{10} + -(1177592394701296536965 + O(13^20))*a - 3977164628725001028259 + O(13^20)q^{11} + -(8158551407268608359854 + O(13^20))*a - 6944010523164507118143 + O(13^20)q^{12} + (2906631575255586571970 + O(13^20))*a + 326152819318163766449 + O(13^20)q^{13} + -(6960200860352474494158 + O(13^20))*a - 464989925717657470243*13 + O(13^20)q^{14} + (709811736544723873376*13 + O(13^20))*a - 8018790776037605998866 + O(13^20)q^{15} + -(8190904040628646290123 + O(13^20))*a + 4818473929426668122047 + O(13^20)q^{16} + -(8187779803144707527820 + O(13^20))*a + 7564928197715657707953 + O(13^20)q^{17} + (2281404053059584564678 + O(13^20))*a - 25127213539461629464*13 + O(13^20)q^{18} + -(2652805818812333321781 + O(13^20))*a + 2258093222292059159622 + O(13^20)q^{19} + (4913510987526563813095 + O(13^20))*a - 8751495551390840135247 + O(13^20)q^{20} + (572239007082756901546*13 + O(13^20))*a - 7436632825165734833930 + O(13^20)q^{21} + -(4064576869278099049489 + O(13^20))*a + 3119186254389819782140 + O(13^20)q^{22} + (43271267814950951678*13^2 + O(13^20))*a - 2466755811907446199643 + O(13^20)q^{23} + (3970451548593424143*13^2 + O(13^20))*a + 1519630202791409810001 + O(13^20)q^{24} + 95367431640625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -768285134085983183771 + O(13^20)x + 7674192862009656936472 + O(13^20)$.
The slope of $f_0$ is $1/2$.

\item  Consider
$f = q + z^129q^{2} + z^43q^{3} + z^38q^{4} + 12q^{5} + z^4q^{6} + z^44q^{7} + z^68q^{8} + z^13q^{9} + z^45q^{10} + z^146q^{11} + z^81q^{12} + z^5q^{14} + z^127q^{15} + z^104q^{16} + z^142q^{17} + z^142q^{18} + z^57q^{19} + z^122q^{20} + z^87q^{21} + z^107q^{22} + z^78q^{23} + z^111q^{24} + q^{25} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + z^115q^{2} + z^155q^{3} + z^10q^{4} + 5q^{5} + z^102q^{6} + z^58q^{7} + z^26q^{8} + z^69q^{9} + z^73q^{10} + z^48q^{11} + z^165q^{12} + z^80q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^88)\omega^{8}&0\\0&u(z^80)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (3137076025398198629864 + O(13^20))*a - 5847327514049658064236 + O(13^20)q^{3} + -(768285134085983180861 + O(13^20))*a - 6362015453619850198340 + O(13^20)q^{4} + 9765625q^{5} + (2308648256769732550905 + O(13^20))*a - 747039781059666053835 + O(13^20)q^{6} + (2096593242060881076046 + O(13^20))*a - 8418047796034689543945 + O(13^20)q^{7} + -(4454785667147864680737 + O(13^20))*a - 8640037411865395963302 + O(13^20)q^{8} + (3597117634209207942738 + O(13^20))*a - 8651629180601800073301 + O(13^20)q^{9} + 9765625*aq^{10} + -(7498846361094114374306 + O(13^20))*a + 5604332821613891952091 + O(13^20)q^{11} + (7366717784328672026128 + O(13^20))*a + 3972222772258492526423 + O(13^20)q^{12} + -(377935360980679455966*13 + O(13^20))*a + 500422341392994369885*13 + O(13^20)q^{13} + -(7009896482071852566527 + O(13^20))*a - 1836964407886041728507 + O(13^20)q^{14} + (584479581700203821826 + O(13^20))*a + 532673804861052205926 + O(13^20)q^{15} + -(7115013471215779958827 + O(13^20))*a - 4814969253528803579688 + O(13^20)q^{16} + (6326566897482629201454 + O(13^20))*a - 3717389997005075258076 + O(13^20)q^{17} + (629450579528827891417 + O(13^20))*a + 5224710588537691081336 + O(13^20)q^{18} + (8791340539968633971521 + O(13^20))*a - 5446783639111509805987 + O(13^20)q^{19} + -(4913510987498145844345 + O(13^20))*a - 6107648627670977468604 + O(13^20)q^{20} + -(8275423512695867990132 + O(13^20))*a + 705717360377302954496*13 + O(13^20)q^{21} + (2437408676422935202105 + O(13^20))*a + 9358959256554706022235 + O(13^20)q^{22} + (2688239783282227518293 + O(13^20))*a - 8127304641654319266167 + O(13^20)q^{23} + -(8197392670674795105521 + O(13^20))*a - 6455597542939726826079 + O(13^20)q^{24} + 95367431640625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 768285134085983180861 + O(13^20)x + 6362015453619848101188 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 9q^{3} + 2q^{4} + 8q^{5} + 12q^{6} + 2q^{7} + 2q^{8} + 7q^{9} + 2q^{10} + 12q^{11} + 5q^{12} + 7q^{14} + 7q^{15} + 6q^{16} + q^{17} + 5q^{18} + 9q^{19} + 3q^{20} + 5q^{21} + 3q^{22} + q^{23} + 5q^{24} + 12q^{25} + \cdots \in S_{24}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{3} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 2q^{2} + 9q^{3} + 11q^{4} + 12q^{5} + 5q^{6} + 10q^{7} + 10q^{8} + 7q^{9} + 11q^{10} + 8q^{11} + 8q^{12} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is irreducible; it is the induction of $\omega_2^{131}$. The form
$f_0 = q + 5556620875575284670152 + O(13^20)q^{2} + 9238283260276156424087 + O(13^20)q^{3} + 1113444115782933846771 + O(13^20)q^{4} + 48828125q^{5} + 7935989305103170261924 + O(13^20)q^{6} + -4069000574030330601388 + O(13^20)q^{7} + 1171628736252281708624 + O(13^20)q^{8} + 3999272949549660055943 + O(13^20)q^{9} + -7341377267633823885638 + O(13^20)q^{10} + 8000908883867868159788 + O(13^20)q^{11} + 7132552052703385138992 + O(13^20)q^{12} + -27540609652591807658*13^2 + O(13^20)q^{13} + 5365945308395518530704 + O(13^20)q^{14} + 8256337833495085641600 + O(13^20)q^{15} + 8267523246998781784414 + O(13^20)q^{16} + -4107047099541136295869 + O(13^20)q^{17} + 2161137051689337762162 + O(13^20)q^{18} + -1318367855324606287621 + O(13^20)q^{19} + 2624944713067083846481 + O(13^20)q^{20} + -284238672550134506251 + O(13^20)q^{21} + 4620690358626070341486 + O(13^20)q^{22} + 7406530942783175415915 + O(13^20)q^{23} + -2266761216723468566447 + O(13^20)q^{24} + 2384185791015625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $2$.

\item  Consider
$f = q + 12q^{2} + 6q^{3} + 7q^{4} + 5q^{5} + 7q^{6} + 11q^{7} + q^{9} + 8q^{10} + 8q^{11} + 3q^{12} + 2q^{14} + 4q^{15} + 3q^{16} + 12q^{18} + 10q^{19} + 9q^{20} + q^{21} + 5q^{22} + 12q^{25} + \cdots \in S_{24}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 9q^{2} + 2q^{3} + 8q^{4} + 8q^{5} + 5q^{6} + 6q^{7} + 3q^{9} + 7q^{10} + 6q^{11} + 3q^{12} + q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(1)\omega^{1}&0\\0&u(1)\omega^{10}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -8820213318452672500794 + O(13^20)q^{2} + -9344900042246398709541 + O(13^20)q^{3} + 3031703063370289829268 + O(13^20)q^{4} + -48828125q^{5} + -4543042019055590619055 + O(13^20)q^{6} + 6629563409341601387189 + O(13^20)q^{7} + 502858814083977613316*13 + O(13^20)q^{8} + -1374063142323900544437 + O(13^20)q^{9} + -8446778327796148760308 + O(13^20)q^{10} + -1401377418737575946932 + O(13^20)q^{11} + -1518848010150012371931 + O(13^20)q^{12} + -382920769198010729045*13 + O(13^20)q^{13} + -9073266018160952708956 + O(13^20)q^{14} + 9121371896864844549027 + O(13^20)q^{15} + -9059604011389120720736 + O(13^20)q^{16} + 29166865604482629927*13^2 + O(13^20)q^{17} + 6730260095178245597022 + O(13^20)q^{18} + 363919564844925632982 + O(13^20)q^{19} + 4411238921713414175043 + O(13^20)q^{20} + -3133577635154775789434 + O(13^20)q^{21} + -5157785890030562021969 + O(13^20)q^{22} + 403194846798276346975*13 + O(13^20)q^{23} + 439686534899962398632*13 + O(13^20)q^{24} + 2384185791015625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^100q^{2} + z^91q^{3} + z^110q^{4} + 5q^{5} + z^23q^{6} + z^92q^{7} + z^21q^{8} + 6q^{9} + z^58q^{10} + z^76q^{11} + z^33q^{12} + z^24q^{14} + z^49q^{15} + z^114q^{16} + z^115q^{17} + z^2q^{18} + z^8q^{19} + z^68q^{20} + z^15q^{21} + z^8q^{22} + z^109q^{23} + 9q^{24} + 12q^{25} + \cdots \in S_{24}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^86q^{2} + z^35q^{3} + z^82q^{4} + q^{5} + z^121q^{6} + z^106q^{7} + z^147q^{8} + 5q^{9} + z^86q^{10} + z^146q^{11} + z^117q^{12} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{10}&0\\0&u(11)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(5588173740060664468381 + O(13^20))*a + 8038334126079748995929 + O(13^20)q^{3} + (1575543532242978676449 + O(13^20))*a - 5335928827825482419660 + O(13^20)q^{4} + -48828125q^{5} + -(3921101660051101246587 + O(13^20))*a + 1164530698400942332963 + O(13^20)q^{6} + (4910211264593652835524 + O(13^20))*a - 8590218479878900641159 + O(13^20)q^{7} + -(5290286604155839233045 + O(13^20))*a + 7457351713744050046274 + O(13^20)q^{8} + -(50460910847679574831*13 + O(13^20))*a - 4600656185715835764487 + O(13^20)q^{9} + -48828125*aq^{10} + -(7336647298535474881561 + O(13^20))*a - 8520575535835728608028 + O(13^20)q^{11} + (5686030920839623525329 + O(13^20))*a + 8645279762916755767266 + O(13^20)q^{12} + (41700739326444700069*13^2 + O(13^20))*a - 716299325687200141011*13 + O(13^20)q^{13} + -(7162258075540135413848 + O(13^20))*a + 6861723610770643660849 + O(13^20)q^{14} + (3476448430939270663121 + O(13^20))*a + 16896236732087353205 + O(13^20)q^{15} + (8696384008184048350007 + O(13^20))*a + 275923906225627905*13 + O(13^20)q^{16} + (4146291709543845910384 + O(13^20))*a - 648632023576352356759 + O(13^20)q^{17} + (2275583519248206939219 + O(13^20))*a - 680243047620055579297*13 + O(13^20)q^{18} + (4874408188065808379346 + O(13^20))*a - 5387612182488668407321 + O(13^20)q^{19} + (2497806640809517159009 + O(13^20))*a - 730379193437133456929 + O(13^20)q^{20} + (8813182732470542764857 + O(13^20))*a - 7893560194467883325763 + O(13^20)q^{21} + (8152822500530779621246 + O(13^20))*a + 4516013678862253613545 + O(13^20)q^{22} + (602394723840443283760 + O(13^20))*a - 2196015909510745073524 + O(13^20)q^{23} + -(362267696735467124230*13 + O(13^20))*a - 8569672676481301813510 + O(13^20)q^{24} + 2384185791015625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -1575543532242978676449 + O(13^20)x + 5335928827825474031052 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^124q^{2} + z^7q^{3} + z^86q^{4} + 5q^{5} + z^131q^{6} + z^20q^{7} + z^105q^{8} + 6q^{9} + z^82q^{10} + z^148q^{11} + z^93q^{12} + z^144q^{14} + z^133q^{15} + z^138q^{16} + z^151q^{17} + z^26q^{18} + z^104q^{19} + z^44q^{20} + z^27q^{21} + z^104q^{22} + z^73q^{23} + 9q^{24} + 12q^{25} + \cdots \in S_{24}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^110q^{2} + z^119q^{3} + z^58q^{4} + q^{5} + z^61q^{6} + z^34q^{7} + z^63q^{8} + 5q^{9} + z^110q^{10} + z^50q^{11} + z^9q^{12} + 11q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(6)\omega^{10}&0\\0&u(11)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(5588173740060664468381 + O(13^20))*a + 8038334126079748995929 + O(13^20)q^{3} + (1575543532242978676449 + O(13^20))*a - 5335928827825482419660 + O(13^20)q^{4} + -48828125q^{5} + -(3921101660051101246587 + O(13^20))*a + 1164530698400942332963 + O(13^20)q^{6} + (4910211264593652835524 + O(13^20))*a - 8590218479878900641159 + O(13^20)q^{7} + -(5290286604155839233045 + O(13^20))*a + 7457351713744050046274 + O(13^20)q^{8} + -(50460910847679574831*13 + O(13^20))*a - 4600656185715835764487 + O(13^20)q^{9} + -48828125*aq^{10} + -(7336647298535474881561 + O(13^20))*a - 8520575535835728608028 + O(13^20)q^{11} + (5686030920839623525329 + O(13^20))*a + 8645279762916755767266 + O(13^20)q^{12} + (41700739326444700069*13^2 + O(13^20))*a - 716299325687200141011*13 + O(13^20)q^{13} + -(7162258075540135413848 + O(13^20))*a + 6861723610770643660849 + O(13^20)q^{14} + (3476448430939270663121 + O(13^20))*a + 16896236732087353205 + O(13^20)q^{15} + (8696384008184048350007 + O(13^20))*a + 275923906225627905*13 + O(13^20)q^{16} + (4146291709543845910384 + O(13^20))*a - 648632023576352356759 + O(13^20)q^{17} + (2275583519248206939219 + O(13^20))*a - 680243047620055579297*13 + O(13^20)q^{18} + (4874408188065808379346 + O(13^20))*a - 5387612182488668407321 + O(13^20)q^{19} + (2497806640809517159009 + O(13^20))*a - 730379193437133456929 + O(13^20)q^{20} + (8813182732470542764857 + O(13^20))*a - 7893560194467883325763 + O(13^20)q^{21} + (8152822500530779621246 + O(13^20))*a + 4516013678862253613545 + O(13^20)q^{22} + (602394723840443283760 + O(13^20))*a - 2196015909510745073524 + O(13^20)q^{23} + -(362267696735467124230*13 + O(13^20))*a - 8569672676481301813510 + O(13^20)q^{24} + 2384185791015625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -1575543532242978676449 + O(13^20)x + 5335928827825474031052 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 3q^{3} + 7q^{4} + 12q^{5} + 9q^{6} + 10q^{7} + 2q^{8} + 6q^{9} + 10q^{10} + 3q^{11} + 8q^{12} + 4q^{14} + 10q^{15} + 5q^{16} + 11q^{17} + 5q^{18} + 8q^{19} + 6q^{20} + 4q^{21} + 9q^{22} + 12q^{23} + 6q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 8q^{2} + q^{3} + 5q^{4} + 5q^{5} + 8q^{6} + 7q^{7} + 10q^{8} + 5q^{9} + q^{10} + 5q^{11} + 5q^{12} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{0}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4029624225301874793877 + O(13^20)q^{2} + -5454035615511091695109 + O(13^20)q^{3} + -7904822684897798928885 + O(13^20)q^{4} + -244140625q^{5} + -6286927990675357244782 + O(13^20)q^{6} + -8690356711864411733555 + O(13^20)q^{7} + 8493843782681485983341 + O(13^20)q^{8} + -4429716151182386651204 + O(13^20)q^{9} + 5284395736852232418766 + O(13^20)q^{10} + -5819814160387020461717 + O(13^20)q^{11} + -8504347407260693384401 + O(13^20)q^{12} + -444812471269482632063*13 + O(13^20)q^{13} + 3193379884152866087489 + O(13^20)q^{14} + 7186777662946109394163 + O(13^20)q^{15} + -2497576911307946566093 + O(13^20)q^{16} + 3532538172932740805741 + O(13^20)q^{17} + -4022197183744891757965 + O(13^20)q^{18} + 3084360946839732161298 + O(13^20)q^{19} + -1369284953583467723017 + O(13^20)q^{20} + 4611384805646377984983 + O(13^20)q^{21} + -5894917916848539085329 + O(13^20)q^{22} + 1587895890471923082764 + O(13^20)q^{23} + 3112114168494502055787 + O(13^20)q^{24} + 59604644775390625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^85q^{2} + z^23q^{3} + z^36q^{4} + q^{5} + z^108q^{6} + z^81q^{7} + z^6q^{8} + z^15q^{9} + z^85q^{10} + z^21q^{11} + z^59q^{12} + z^166q^{14} + z^23q^{15} + z^30q^{16} + z^147q^{17} + z^100q^{18} + z^36q^{20} + z^104q^{21} + z^106q^{22} + z^82q^{23} + z^29q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^71q^{2} + z^135q^{3} + z^8q^{4} + 8q^{5} + z^38q^{6} + z^95q^{7} + z^132q^{8} + z^71q^{9} + z^113q^{10} + z^91q^{11} + z^143q^{12} + z^123q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^45)\omega^{0}&0\\0&u(z^123)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (4817024871679759910491 + O(13^20))*a - 5238730844434464704939 + O(13^20)q^{3} + (3225649235121353490722 + O(13^20))*a - 6471259024495946884285 + O(13^20)q^{4} + 244140625q^{5} + -(6493714410467951448895 + O(13^20))*a - 3347797951523560593204 + O(13^20)q^{6} + -(2497547174853538828288 + O(13^20))*a - 6564615942529186529024 + O(13^20)q^{7} + (4009325613210611031034 + O(13^20))*a + 568654097995765401388 + O(13^20)q^{8} + (2732960379137993655558 + O(13^20))*a + 701135667072764056141*13 + O(13^20)q^{9} + 244140625*aq^{10} + -(7224751876456490397373 + O(13^20))*a + 3259318721005238008929 + O(13^20)q^{11} + -(79211798230931188664 + O(13^20))*a + 9319433509915452225959 + O(13^20)q^{12} + -(485552150180362342871*13 + O(13^20))*a + 464792176838410971625*13 + O(13^20)q^{13} + (7928020701746861218357 + O(13^20))*a - 5639968579109909497905 + O(13^20)q^{14} + (9301004392063537747849 + O(13^20))*a + 2197126987648448220650 + O(13^20)q^{15} + (8375979133029538977722 + O(13^20))*a + 8920441091876197701693 + O(13^20)q^{16} + (434277189605808982233 + O(13^20))*a + 4345445697441727193072 + O(13^20)q^{17} + -(6298916763814019474920 + O(13^20))*a - 7421195438995738762267 + O(13^20)q^{18} + -(585319522320448659722*13 + O(13^20))*a - 582026080329355693541*13 + O(13^20)q^{19} + -(8520060873295497484726 + O(13^20))*a - 7364274591375445887758 + O(13^20)q^{20} + (2824734236630031092624 + O(13^20))*a + 9029005023699013092762 + O(13^20)q^{21} + (7908832809482419460952 + O(13^20))*a - 3215224639950137163490 + O(13^20)q^{22} + (2185984208655910488374 + O(13^20))*a + 3053387386610046729409 + O(13^20)q^{23} + (4124987653296844901784 + O(13^20))*a - 4418910412511518567*13^2 + O(13^20)q^{24} + 59604644775390625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3225649235121353490722 + O(13^20)x + 6471259024495913329853 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^97q^{2} + z^131q^{3} + z^132q^{4} + q^{5} + z^60q^{6} + z^45q^{7} + z^78q^{8} + z^27q^{9} + z^97q^{10} + z^105q^{11} + z^95q^{12} + z^142q^{14} + z^131q^{15} + z^54q^{16} + z^63q^{17} + z^124q^{18} + z^132q^{20} + z^8q^{21} + z^34q^{22} + z^58q^{23} + z^41q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^83q^{2} + z^75q^{3} + z^104q^{4} + 8q^{5} + z^158q^{6} + z^59q^{7} + z^36q^{8} + z^83q^{9} + z^125q^{10} + z^7q^{11} + z^11q^{12} + z^87q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^81)\omega^{0}&0\\0&u(z^87)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (4817024871679759910491 + O(13^20))*a - 5238730844434464704939 + O(13^20)q^{3} + (3225649235121353490722 + O(13^20))*a - 6471259024495946884285 + O(13^20)q^{4} + 244140625q^{5} + -(6493714410467951448895 + O(13^20))*a - 3347797951523560593204 + O(13^20)q^{6} + -(2497547174853538828288 + O(13^20))*a - 6564615942529186529024 + O(13^20)q^{7} + (4009325613210611031034 + O(13^20))*a + 568654097995765401388 + O(13^20)q^{8} + (2732960379137993655558 + O(13^20))*a + 701135667072764056141*13 + O(13^20)q^{9} + 244140625*aq^{10} + -(7224751876456490397373 + O(13^20))*a + 3259318721005238008929 + O(13^20)q^{11} + -(79211798230931188664 + O(13^20))*a + 9319433509915452225959 + O(13^20)q^{12} + -(485552150180362342871*13 + O(13^20))*a + 464792176838410971625*13 + O(13^20)q^{13} + (7928020701746861218357 + O(13^20))*a - 5639968579109909497905 + O(13^20)q^{14} + (9301004392063537747849 + O(13^20))*a + 2197126987648448220650 + O(13^20)q^{15} + (8375979133029538977722 + O(13^20))*a + 8920441091876197701693 + O(13^20)q^{16} + (434277189605808982233 + O(13^20))*a + 4345445697441727193072 + O(13^20)q^{17} + -(6298916763814019474920 + O(13^20))*a - 7421195438995738762267 + O(13^20)q^{18} + -(585319522320448659722*13 + O(13^20))*a - 582026080329355693541*13 + O(13^20)q^{19} + -(8520060873295497484726 + O(13^20))*a - 7364274591375445887758 + O(13^20)q^{20} + (2824734236630031092624 + O(13^20))*a + 9029005023699013092762 + O(13^20)q^{21} + (7908832809482419460952 + O(13^20))*a - 3215224639950137163490 + O(13^20)q^{22} + (2185984208655910488374 + O(13^20))*a + 3053387386610046729409 + O(13^20)q^{23} + (4124987653296844901784 + O(13^20))*a - 4418910412511518567*13^2 + O(13^20)q^{24} + 59604644775390625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + -3225649235121353490722 + O(13^20)x + 6471259024495913329853 + O(13^20)$.
The slope of $f_0$ is $1$.

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(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 9q^{2} + 2q^{3} + 8q^{4} + 8q^{5} + 5q^{6} + 6q^{7} + 3q^{9} + 7q^{10} + 6q^{11} + 3q^{12} + q^{13} + 2q^{14} + 3q^{15} + q^{16} + q^{18} + 9q^{19} + 12q^{20} + 12q^{21} + 2q^{22} + 12q^{25} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 12q^{3} + 2q^{4} + 5q^{5} + 7q^{6} + 4q^{7} + 3q^{8} + 4q^{10} + 2q^{11} + 11q^{12} + 11q^{14} + 8q^{15} + 2q^{16} + 7q^{17} + 3q^{19} + 10q^{20} + 9q^{21} + 12q^{22} + 6q^{23} + 10q^{24} + 12q^{25} + \cdots \in S_{16}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\item  Consider
$f = q + z^115q^{2} + z^155q^{3} + z^10q^{4} + 5q^{5} + z^102q^{6} + z^58q^{7} + z^26q^{8} + z^69q^{9} + z^73q^{10} + z^48q^{11} + z^165q^{12} + z^80q^{13} + z^5q^{14} + z^113q^{15} + z^48q^{16} + z^114q^{17} + z^16q^{18} + z^155q^{19} + z^136q^{20} + z^45q^{21} + z^163q^{22} + z^106q^{23} + z^13q^{24} + 12q^{25} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^151q^{2} + z^167q^{3} + z^130q^{4} + 5q^{5} + z^150q^{6} + z^82q^{7} + z^2q^{8} + z^57q^{9} + z^109q^{10} + z^120q^{11} + z^129q^{12} + z^32q^{13} + z^65q^{14} + z^125q^{15} + z^120q^{16} + z^138q^{17} + z^40q^{18} + z^167q^{19} + z^88q^{20} + z^81q^{21} + z^103q^{22} + z^34q^{23} + zq^{24} + 12q^{25} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 12q^{2} + 4q^{3} + 3q^{4} + 8q^{5} + 9q^{6} + 7q^{7} + 8q^{8} + 5q^{10} + 3q^{11} + 12q^{12} + q^{13} + 6q^{14} + 6q^{15} + 11q^{16} + 5q^{17} + 8q^{19} + 11q^{20} + 2q^{21} + 10q^{22} + 11q^{23} + 6q^{24} + 12q^{25} + \cdots \in S_{20}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^86q^{2} + z^35q^{3} + z^82q^{4} + q^{5} + z^121q^{6} + z^106q^{7} + z^147q^{8} + 5q^{9} + z^86q^{10} + z^146q^{11} + z^117q^{12} + 11q^{13} + z^24q^{14} + z^35q^{15} + z^58q^{16} + z^87q^{17} + z^44q^{18} + z^106q^{19} + z^82q^{20} + z^141q^{21} + z^64q^{22} + z^137q^{23} + 2q^{24} + q^{25} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^110q^{2} + z^119q^{3} + z^58q^{4} + q^{5} + z^61q^{6} + z^34q^{7} + z^63q^{8} + 5q^{9} + z^110q^{10} + z^50q^{11} + z^9q^{12} + 11q^{13} + z^144q^{14} + z^119q^{15} + z^82q^{16} + z^123q^{17} + z^68q^{18} + z^34q^{19} + z^58q^{20} + z^153q^{21} + z^160q^{22} + z^101q^{23} + 2q^{24} + q^{25} + \cdots \in S_{22}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^71q^{2} + z^135q^{3} + z^8q^{4} + 8q^{5} + z^38q^{6} + z^95q^{7} + z^132q^{8} + z^71q^{9} + z^113q^{10} + z^91q^{11} + z^143q^{12} + z^123q^{13} + z^166q^{14} + z^9q^{15} + z^142q^{16} + z^119q^{17} + z^142q^{18} + z^50q^{20} + z^62q^{21} + z^162q^{22} + z^110q^{23} + z^99q^{24} + 12q^{25} + \cdots \in S_{24}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^83q^{2} + z^75q^{3} + z^104q^{4} + 8q^{5} + z^158q^{6} + z^59q^{7} + z^36q^{8} + z^83q^{9} + z^125q^{10} + z^7q^{11} + z^11q^{12} + z^87q^{13} + z^142q^{14} + z^117q^{15} + z^166q^{16} + z^35q^{17} + z^166q^{18} + z^146q^{20} + z^134q^{21} + z^90q^{22} + z^86q^{23} + z^111q^{24} + 12q^{25} + \cdots \in S_{24}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + q^{3} + 5q^{4} + 5q^{5} + 8q^{6} + 7q^{7} + 10q^{8} + 5q^{9} + q^{10} + 5q^{11} + 5q^{12} + 7q^{13} + 4q^{14} + 5q^{15} + 6q^{16} + 6q^{17} + q^{18} + 10q^{19} + 12q^{20} + 7q^{21} + q^{22} + 9q^{23} + 10q^{24} + 12q^{25} + \cdots \in S_{24}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 12q^{2} + 11q^{3} + 12q^{4} + 12q^{5} + 2q^{6} + 9q^{7} + 3q^{8} + q^{9} + q^{10} + 2q^{11} + 2q^{12} + 12q^{13} + 4q^{14} + 2q^{15} + 12q^{16} + 2q^{17} + 12q^{18} + 7q^{19} + q^{20} + 8q^{21} + 11q^{22} + 7q^{23} + 7q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 9q^{2} + 5q^{3} + q^{4} + 12q^{5} + 6q^{6} + 2q^{7} + 4q^{8} + 9q^{9} + 4q^{10} + 6q^{11} + 5q^{12} + q^{13} + 5q^{14} + 8q^{15} + 8q^{16} + 5q^{17} + 3q^{18} + 12q^{20} + 10q^{21} + 2q^{22} + 12q^{23} + 7q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 10q^{3} + q^{4} + 12q^{5} + q^{6} + 2q^{7} + 9q^{8} + 6q^{9} + 9q^{10} + q^{11} + 10q^{12} + q^{13} + 8q^{14} + 3q^{15} + 8q^{16} + 8q^{17} + 11q^{18} + 11q^{19} + 12q^{20} + 7q^{21} + 4q^{22} + 7q^{23} + 12q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^102q^{2} + z^91q^{3} + z^89q^{4} + q^{5} + z^25q^{6} + z^32q^{7} + z^167q^{8} + 12q^{9} + z^102q^{10} + z^25q^{11} + z^12q^{12} + 12q^{13} + z^134q^{14} + z^91q^{15} + 3q^{16} + z^164q^{17} + z^18q^{18} + z^157q^{19} + z^89q^{20} + z^123q^{21} + z^127q^{22} + z^7q^{23} + z^90q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + z^150q^{2} + z^7q^{3} + z^149q^{4} + q^{5} + z^157q^{6} + z^80q^{7} + z^155q^{8} + 12q^{9} + z^150q^{10} + z^157q^{11} + z^156q^{12} + 12q^{13} + z^62q^{14} + z^7q^{15} + 3q^{16} + z^116q^{17} + z^66q^{18} + z^25q^{19} + z^149q^{20} + z^87q^{21} + z^139q^{22} + z^91q^{23} + z^162q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 3q^{2} + 4q^{3} + 7q^{4} + q^{5} + 12q^{6} + 8q^{7} + 2q^{8} + 3q^{10} + 12q^{11} + 2q^{12} + 11q^{14} + 4q^{15} + 5q^{16} + 5q^{17} + 7q^{19} + 7q^{20} + 6q^{21} + 10q^{22} + 11q^{23} + 8q^{24} + q^{25} + \cdots \in S_{26}(\Gamma_0(5);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
I didn't find a twist of $\rho_f$ in low weight, so $\rho_f$ must be reducible and
we do not consider it further.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 6, \quad \ell = 13}$ \quad (20 forms)}

\begin{enumerate}
\item  Consider
$f = q + 11q^{2} + 3q^{3} + 4q^{4} + 10q^{5} + 7q^{6} + 5q^{7} + 5q^{8} + 9q^{9} + 6q^{10} + q^{11} + 12q^{12} + 3q^{14} + 4q^{15} + 3q^{16} + 8q^{18} + 6q^{19} + q^{20} + 2q^{21} + 11q^{22} + 2q^{24} + q^{25} + \cdots \in S_{16}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 6q^{2} + 9q^{3} + 10q^{4} + 3q^{5} + 2q^{6} + 7q^{7} + 8q^{8} + 3q^{9} + 5q^{10} + 10q^{11} + 12q^{12} + 12q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(12)\omega^{1}&0\\0&u(12)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 128q^{2} + 2187q^{3} + 16384q^{4} + 77646q^{5} + 279936q^{6} + 762104q^{7} + 2097152q^{8} + 4782969q^{9} + 9938688q^{10} + 48011172q^{11} + 35831808q^{12} + 21933086*13q^{13} + 97549312q^{14} + 169811802q^{15} + 268435456q^{16} + -244128582*13q^{17} + 612220032q^{18} + -5895116260q^{19} + 1272152064q^{20} + 1666721448q^{21} + 6145430016q^{22} + -25616184*13q^{23} + 4586471424q^{24} + -24488676809q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 2q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 7q^{6} + 7q^{7} + 8q^{8} + 9q^{9} + 12q^{10} + 11q^{11} + q^{12} + q^{14} + 8q^{15} + 3q^{16} + 9q^{17} + 5q^{18} + 10q^{19} + 11q^{20} + 5q^{21} + 9q^{22} + 6q^{23} + 2q^{24} + 2q^{25} + \cdots \in S_{16}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 7q^{2} + 4q^{3} + 10q^{4} + 7q^{5} + 2q^{6} + 2q^{7} + 5q^{8} + 3q^{9} + 10q^{10} + 6q^{11} + q^{12} + 10q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{1}&0\\0&u(10)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -128q^{2} + -2187q^{3} + 16384q^{4} + -314490q^{5} + 279936q^{6} + 2025056q^{7} + -2097152q^{8} + 4782969q^{9} + 40254720q^{10} + 110255052q^{11} + -35831808q^{12} + 4311374*13q^{13} + -259207168q^{14} + 687789630q^{15} + 268435456q^{16} + -1930104414q^{17} + -612220032q^{18} + 2163188180q^{19} + -5152604160q^{20} + -4428797472q^{21} + -14112646656q^{22} + 6228974472q^{23} + 4586471424q^{24} + 68386381975q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 4q^{2} + 9q^{3} + 3q^{4} + 3q^{5} + 10q^{6} + 6q^{7} + 12q^{8} + 3q^{9} + 12q^{10} + 2q^{11} + q^{12} + 11q^{14} + q^{15} + 9q^{16} + 4q^{17} + 12q^{18} + 6q^{19} + 9q^{20} + 2q^{21} + 8q^{22} + 11q^{23} + 4q^{24} + 4q^{25} + \cdots \in S_{18}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 10q^{2} + 3q^{3} + 9q^{4} + 3q^{5} + 4q^{6} + 5q^{7} + 12q^{8} + 9q^{9} + 4q^{10} + 5q^{11} + q^{12} + 6q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(11)\omega^{1}&0\\0&u(6)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -256q^{2} + 6561q^{3} + 65536q^{4} + -72186q^{5} + -1679616q^{6} + -8640184q^{7} + -16777216q^{8} + 43046721q^{9} + 18479616q^{10} + 1159304460q^{11} + 429981696q^{12} + 215466374*13q^{13} + 2211887104q^{14} + -473612346q^{15} + 4294967296q^{16} + 32979662226q^{17} + -11019960576q^{18} + 5778498836q^{19} + -4730781696q^{20} + -56688247224q^{21} + -296781941760q^{22} + 169116994200q^{23} + -110075314176q^{24} + -757728634529q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 9q^{2} + 4q^{3} + 3q^{4} + 4q^{5} + 10q^{6} + 5q^{7} + q^{8} + 3q^{9} + 10q^{10} + 2q^{11} + 12q^{12} + 6q^{14} + 3q^{15} + 9q^{16} + 3q^{17} + q^{18} + 3q^{19} + 12q^{20} + 7q^{21} + 5q^{22} + 3q^{23} + 4q^{24} + 11q^{25} + \cdots \in S_{18}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 11q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 10q^{7} + 5q^{8} + 9q^{9} + q^{10} + 12q^{11} + q^{12} + 12q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(12)\omega^{4}&0\\0&u(12)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 256q^{2} + -6561q^{3} + 65536q^{4} + -199650q^{5} + -1679616q^{6} + 24959264q^{7} + 16777216q^{8} + 43046721q^{9} + -51110400q^{10} + 125556420q^{11} + -429981696q^{12} + 325168886*13q^{13} + 6389571584q^{14} + 1309903650q^{15} + 4294967296q^{16} + 35551782594q^{17} + 11019960576q^{18} + -64354589764q^{19} + -13084262400q^{20} + -163757731104q^{21} + 32142443520q^{22} + -245819296200q^{23} + -110075314176q^{24} + -723079330625q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 5q^{2} + q^{3} + 12q^{4} + 10q^{5} + 5q^{6} + 3q^{7} + 8q^{8} + q^{9} + 11q^{10} + 12q^{12} + 2q^{14} + 10q^{15} + q^{16} + 11q^{17} + 5q^{18} + 9q^{19} + 3q^{20} + 3q^{21} + 4q^{23} + 8q^{24} + q^{25} + \cdots \in S_{20}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{6} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + q^{3} + 12q^{4} + 3q^{5} + 8q^{6} + 10q^{7} + 5q^{8} + q^{9} + 11q^{10} + 12q^{12} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{1}&0\\0&u(7)\omega^{6}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 512q^{2} + 19683q^{3} + 262144q^{4} + 1953390q^{5} + 10077696q^{6} + 40488776q^{7} + 134217728q^{8} + 387420489q^{9} + 1000135680q^{10} + 147450804*13q^{11} + 5159780352q^{12} + 240960014*13q^{13} + 20730253312q^{14} + 38448575370q^{15} + 68719476736q^{16} + 607659965586q^{17} + 198359290368q^{18} + 2507511106460q^{19} + 512069468160q^{20} + 796940578008q^{21} + 75494811648*13q^{22} + -13588841327928q^{23} + 2641807540224q^{24} + -15257753836025q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 8q^{2} + 12q^{3} + 12q^{4} + 8q^{5} + 5q^{6} + 10q^{7} + 5q^{8} + q^{9} + 12q^{10} + 3q^{11} + q^{12} + 2q^{14} + 5q^{15} + q^{16} + 8q^{17} + 8q^{18} + 3q^{19} + 5q^{20} + 3q^{21} + 11q^{22} + 7q^{23} + 8q^{24} + 4q^{25} + \cdots \in S_{20}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 4q^{3} + 3q^{4} + 12q^{5} + 3q^{6} + 7q^{7} + 12q^{8} + 3q^{9} + 9q^{10} + 5q^{11} + 12q^{12} + 5q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{6}&0\\0&u(5)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -512q^{2} + -19683q^{3} + 262144q^{4} + -3732474q^{5} + 10077696q^{6} + -149672656q^{7} + -134217728q^{8} + 387420489q^{9} + 1911026688q^{10} + -7459672308q^{11} + -5159780352q^{12} + 4556804606*13q^{13} + 76632399872q^{14} + 73466285742q^{15} + 68719476736q^{16} + 523110429954q^{17} + -198359290368q^{18} + 969502037780q^{19} + -978445664256q^{20} + 2946006888048q^{21} + 3819352221696q^{22} + -1368374071512q^{23} + 2641807540224q^{24} + -5142124167449q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 10q^{2} + 10q^{3} + 9q^{4} + 12q^{5} + 9q^{6} + 8q^{7} + 12q^{8} + 9q^{9} + 3q^{10} + 6q^{11} + 12q^{12} + 2q^{14} + 3q^{15} + 3q^{16} + 6q^{17} + 12q^{18} + 8q^{19} + 4q^{20} + 2q^{21} + 8q^{22} + 5q^{23} + 3q^{24} + 9q^{25} + \cdots \in S_{22}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{4} \sim \rho_g$, where $g$ is the
weight-$6$ newform
$g = q + 4q^{2} + 4q^{3} + 3q^{4} + 12q^{5} + 3q^{6} + 7q^{7} + 12q^{8} + 3q^{9} + 9q^{10} + 5q^{11} + 12q^{12} + 5q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(8)\omega^{1}&0\\0&u(5)\omega^{8}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 1024q^{2} + -59049q^{3} + 1048576q^{4} + 12954174q^{5} + -60466176q^{6} + -479513104q^{7} + 1073741824q^{8} + 3486784401q^{9} + 13265074176q^{10} + 115657781700q^{11} + -61917364224q^{12} + 22742942054*13q^{13} + -491021418496q^{14} + -764931020526q^{15} + 1099511627776q^{16} + 6626983431906q^{17} + 3570467226624q^{18} + 28576184164796q^{19} + 13583435956224q^{20} + 28314769278096q^{21} + 118433568460800q^{22} + 335385196791000q^{23} + -63403380965376q^{24} + -309026534180849q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 3q^{2} + 3q^{3} + 9q^{4} + 2q^{5} + 9q^{6} + 5q^{7} + q^{8} + 9q^{9} + 6q^{10} + q^{12} + 2q^{14} + 6q^{15} + 3q^{16} + 5q^{17} + q^{18} + 11q^{19} + 5q^{20} + 2q^{21} + q^{23} + 3q^{24} + 12q^{25} + \cdots \in S_{22}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$8$ newform
$g = q + 8q^{2} + q^{3} + 12q^{4} + 3q^{5} + 8q^{6} + 10q^{7} + 5q^{8} + q^{9} + 11q^{10} + 12q^{12} + 7q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(2)\omega^{8}&0\\0&u(7)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -1024q^{2} + 59049q^{3} + 1048576q^{4} + 26444550q^{5} + -60466176q^{6} + 166115864q^{7} + -1073741824q^{8} + 3486784401q^{9} + -27079219200q^{10} + -8067597060*13q^{11} + 61917364224q^{12} + 25814717366*13q^{13} + -170102644736q^{14} + 1561524232950q^{15} + 1099511627776q^{16} + 14596144763634q^{17} + -3570467226624q^{18} + 3569529974996q^{19} + 27729120460800q^{20} + 9808975653336q^{21} + 8261219389440*13q^{22} + 222369240588600q^{23} + -63403380965376q^{24} + 222477066499375q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 6q^{2} + 4q^{3} + 10q^{4} + 7q^{5} + 11q^{6} + q^{7} + 8q^{8} + 3q^{9} + 3q^{10} + 3q^{11} + q^{12} + 6q^{14} + 2q^{15} + 9q^{16} + 10q^{17} + 5q^{18} + 2q^{19} + 5q^{20} + 4q^{21} + 5q^{22} + 10q^{23} + 6q^{24} + 2q^{25} + \cdots \in S_{24}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{2} \sim \rho_g$, where $g$ is the
weight-$4$ newform
$g = q + 11q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 10q^{7} + 5q^{8} + 9q^{9} + q^{10} + 12q^{11} + q^{12} + 12q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(12)\omega^{1}&0\\0&u(12)\omega^{10}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -2048q^{2} + -177147q^{3} + 4194304q^{4} + 196251270q^{5} + 362797056q^{6} + -8131131904q^{7} + -8589934592q^{8} + 31381059609q^{9} + -401922600960q^{10} + 413280901452q^{11} + -743008370688q^{12} + -455779544146*13q^{13} + 16652558139392q^{14} + -34765323726690q^{15} + 17592186044416q^{16} + 44831041228386q^{17} + -64268410079232q^{18} + 869707616627540q^{19} + 823137486766080q^{20} + 1440405623397888q^{21} + -846399286173696q^{22} + -3916239181103928q^{23} + 1521681143169024q^{24} + 26593632021534775q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 7q^{2} + 9q^{3} + 10q^{4} + 2q^{5} + 11q^{6} + 9q^{7} + 5q^{8} + 3q^{9} + q^{10} + 3q^{11} + 12q^{12} + 11q^{14} + 5q^{15} + 9q^{16} + 9q^{17} + 8q^{18} + 4q^{19} + 7q^{20} + 3q^{21} + 8q^{22} + 2q^{23} + 6q^{24} + 9q^{25} + \cdots \in S_{24}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + 10q^{2} + 3q^{3} + 9q^{4} + 3q^{5} + 4q^{6} + 5q^{7} + 12q^{8} + 9q^{9} + 4q^{10} + 5q^{11} + q^{12} + 6q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(11)\omega^{10}&0\\0&u(6)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 2048q^{2} + 177147q^{3} + 4194304q^{4} + -678040920585597483315 + O(13^20)q^{5} + 362797056q^{6} + -2053940760237664522592 + O(13^20)q^{7} + 8589934592q^{8} + 31381059609q^{9} + -1265449793005286796647 + O(13^20)q^{10} + -2924605830802968318354 + O(13^20)q^{11} + 743008370688q^{12} + 324506150306473008911*13 + O(13^20)q^{13} + -6373682718080266293395 + O(13^20)q^{14} + -1543901730184923579985 + O(13^20)q^{15} + 17592186044416q^{16} + -1249174841287578647173 + O(13^20)q^{17} + 64268410079232q^{18} + 1078342756058768380639 + O(13^20)q^{19} + -6966102691038635856120 + O(13^20)q^{20} + 587616271348072240121 + O(13^20)q^{21} + -3029152397027292766677 + O(13^20)q^{22} + 9144697135906431425211 + O(13^20)q^{23} + 1521681143169024q^{24} + 8331831115340023375015 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + 12q^{2} + q^{3} + q^{4} + 2q^{5} + 12q^{6} + 10q^{7} + 12q^{8} + q^{9} + 11q^{10} + 6q^{11} + q^{12} + 3q^{14} + 2q^{15} + q^{16} + 12q^{18} + q^{19} + 2q^{20} + 10q^{21} + 7q^{22} + 12q^{24} + 12q^{25} + \cdots \in S_{26}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 6q^{2} + 9q^{3} + 10q^{4} + 3q^{5} + 2q^{6} + 7q^{7} + 8q^{8} + 3q^{9} + 5q^{10} + 10q^{11} + 12q^{12} + 12q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(12)\omega^{0}&0\\0&u(12)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + -4096q^{2} + 531441q^{3} + 16777216q^{4} + 590425734q^{5} + -2176782336q^{6} + 57857417576q^{7} + -68719476736q^{8} + 282429536481q^{9} + -2418383806464q^{10} + 9494266240140q^{11} + 8916100448256q^{12} + -10382155466266*13q^{13} + -236983982391296q^{14} + 313776442502694q^{15} + 281474976710656q^{16} + -194316462831078*13q^{17} + -1156831381426176q^{18} + 11468758872260756q^{19} + 9905700071276544q^{20} + 30747803854007016q^{21} + -38888514519613440q^{22} + 8718663907846200*13q^{23} + -36520347436056576q^{24} + 50579323492485631q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + q^{2} + 12q^{3} + q^{4} + 9q^{5} + 12q^{6} + q^{7} + q^{8} + q^{9} + 9q^{10} + q^{11} + 12q^{12} + q^{14} + 4q^{15} + q^{16} + 12q^{17} + q^{18} + 6q^{19} + 9q^{20} + 12q^{21} + q^{22} + 11q^{23} + 12q^{24} + 11q^{25} + \cdots \in S_{26}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
We have $\rho_f \otimes \chi^{11} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + 7q^{2} + 4q^{3} + 10q^{4} + 7q^{5} + 2q^{6} + 2q^{7} + 5q^{8} + 3q^{9} + 10q^{10} + 6q^{11} + q^{12} + 10q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(4)\omega^{0}&0\\0&u(10)\omega^{1}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + 4096q^{2} + -531441q^{3} + 16777216q^{4} + -799327650q^{5} + -2176782336q^{6} + 7962409664q^{7} + 68719476736q^{8} + 282429536481q^{9} + -3274046054400q^{10} + -1760169438780q^{11} + -8916100448256q^{12} + 12630597778646*13q^{13} + 32614029983744q^{14} + 424795485643650q^{15} + 281474976710656q^{16} + -2377485783158526q^{17} + 1156831381426176q^{18} + 10628455411412156q^{19} + -13410492638822400q^{20} + -4231550954245824q^{21} + -7209654021242880q^{22} + 76636270038478200q^{23} + -36520347436056576q^{24} + 340901468177569375q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
The slope of $f_0$ is $1$.

----------------------------------------------------------------------------------------------------\\
(Uninteresting cases below.)
\begin{tiny}
\item  Consider
$f = q + 11q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 10q^{7} + 5q^{8} + 9q^{9} + q^{10} + 12q^{11} + q^{12} + 12q^{13} + 6q^{14} + 8q^{15} + 3q^{16} + 4q^{17} + 8q^{18} + 7q^{19} + 11q^{20} + 9q^{21} + 2q^{22} + 12q^{23} + 11q^{24} + 2q^{25} + \cdots \in S_{16}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 4q^{2} + 4q^{3} + 3q^{4} + 12q^{5} + 3q^{6} + 7q^{7} + 12q^{8} + 3q^{9} + 9q^{10} + 5q^{11} + 12q^{12} + 5q^{13} + 2q^{14} + 9q^{15} + 9q^{16} + 2q^{17} + 12q^{18} + 7q^{19} + 10q^{20} + 2q^{21} + 7q^{22} + 2q^{23} + 9q^{24} + 9q^{25} + \cdots \in S_{18}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 8q^{2} + q^{3} + 12q^{4} + 3q^{5} + 8q^{6} + 10q^{7} + 5q^{8} + q^{9} + 11q^{10} + 12q^{12} + 7q^{13} + 2q^{14} + 3q^{15} + q^{16} + 11q^{17} + 8q^{18} + 4q^{19} + 10q^{20} + 10q^{21} + 4q^{23} + 5q^{24} + q^{25} + \cdots \in S_{20}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 10q^{2} + 3q^{3} + 9q^{4} + 3q^{5} + 4q^{6} + 5q^{7} + 12q^{8} + 9q^{9} + 4q^{10} + 5q^{11} + q^{12} + 6q^{13} + 11q^{14} + 9q^{15} + 3q^{16} + 12q^{17} + 12q^{18} + 5q^{19} + q^{20} + 2q^{21} + 11q^{22} + 8q^{23} + 10q^{24} + 4q^{25} + \cdots \in S_{22}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 4q^{3} + 10q^{4} + 7q^{5} + 2q^{6} + 2q^{7} + 5q^{8} + 3q^{9} + 10q^{10} + 6q^{11} + q^{12} + 10q^{13} + q^{14} + 2q^{15} + 9q^{16} + 3q^{17} + 8q^{18} + q^{19} + 5q^{20} + 8q^{21} + 3q^{22} + 5q^{23} + 7q^{24} + 2q^{25} + \cdots \in S_{24}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 6q^{2} + 9q^{3} + 10q^{4} + 3q^{5} + 2q^{6} + 7q^{7} + 8q^{8} + 3q^{9} + 5q^{10} + 10q^{11} + 12q^{12} + 12q^{13} + 3q^{14} + q^{15} + 9q^{16} + 5q^{18} + 11q^{19} + 4q^{20} + 11q^{21} + 8q^{22} + 7q^{24} + q^{25} + \cdots \in S_{24}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 7q^{2} + 9q^{3} + 10q^{4} + 9q^{5} + 11q^{6} + 3q^{7} + 5q^{8} + 3q^{9} + 11q^{10} + 7q^{11} + 12q^{12} + q^{13} + 8q^{14} + 3q^{15} + 9q^{16} + 11q^{17} + 8q^{18} + 12q^{19} + 12q^{20} + q^{21} + 10q^{22} + 5q^{23} + 6q^{24} + 8q^{25} + \cdots \in S_{24}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\item  Consider
$f = q + 12q^{2} + 12q^{3} + q^{4} + 2q^{5} + q^{6} + 4q^{7} + 12q^{8} + q^{9} + 11q^{10} + 9q^{11} + 12q^{12} + q^{13} + 9q^{14} + 11q^{15} + q^{16} + 2q^{17} + 12q^{18} + 5q^{19} + 2q^{20} + 9q^{21} + 4q^{22} + q^{24} + 12q^{25} + \cdots \in S_{26}(\Gamma_0(6);\overline{\mathbf{F}}_{13}).$
Since~$f$ is ordinary, we do not continue our analysis.

\end{tiny}
\end{enumerate}

\subsection{$\mathbf{N = 7, \quad \ell = 13}$ \quad (56 forms)}

\begin{enumerate}
\item  Consider
$f = q + z^130q^{2} + zq^{3} + z^143q^{4} + z^122q^{5} + z^131q^{6} + 6q^{7} + z^50q^{8} + z^39q^{9} + 12q^{10} + 11q^{11} + z^144q^{12} + z^32q^{14} + z^123q^{15} + z^44q^{16} + z^133q^{17} + zq^{18} + z^97q^{19} + z^97q^{20} + z^71q^{21} + z^60q^{22} + z^164q^{23} + z^51q^{24} + z^138q^{25} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^102q^{2} + z^57q^{3} + z^87q^{4} + z^38q^{5} + z^159q^{6} + 11q^{7} + z^134q^{8} + z^151q^{9} + 10q^{10} + 6q^{11} + z^144q^{12} + z^167q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z)\omega^{1}&0\\0&u(z^167)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(9329958858288907968487 + O(13^20))*a + 7562562216112441623311 + O(13^20)q^{3} + -(2760368466423868014502 + O(13^20))*a - 6971212808514269314810 + O(13^20)q^{4} + (8267390115163250621502 + O(13^20))*a - 259606510531290228534 + O(13^20)q^{5} + -(207893217219449301505 + O(13^20))*a + 6557687179708102195828 + O(13^20)q^{6} + 823543q^{7} + -(7276293298716973815740 + O(13^20))*a - 6407957626558759662389 + O(13^20)q^{8} + -(4612632675558796866387 + O(13^20))*a + 3657550997986767747634 + O(13^20)q^{9} + -(127045723787799152879*13 + O(13^20))*a - 1313135677938326435025 + O(13^20)q^{10} + -(35477406620544946425*13^2 + O(13^20))*a + 511221811079119917931 + O(13^20)q^{11} + -(2870244219471936583651 + O(13^20))*a - 510752989406308633064*13 + O(13^20)q^{12} + (453054560525487782930*13 + O(13^20))*a - 636101102354146201010*13 + O(13^20)q^{13} + 823543*aq^{14} + (4045000997094555040251 + O(13^20))*a + 9091857072255346124734 + O(13^20)q^{15} + (33108491273964405228991 + O(13^21))*a - 104878616586830885839068 + O(13^21)q^{16} + (4002448281428558961755 + O(13^20))*a - 8052215476668073979351 + O(13^20)q^{17} + (4327911188296000936169 + O(13^20))*a + 3606810869568952698385 + O(13^20)q^{18} + -(2420627208219898283522 + O(13^20))*a + 4938669912157861915975 + O(13^20)q^{19} + -(7684293478524480068193 + O(13^20))*a + 1008598992274449635128 + O(13^20)q^{20} + (8526297385554817799655 + O(13^20))*a - 2498558523391503626036 + O(13^20)q^{21} + -(9288199545084343608467 + O(13^20))*a + 6185546384500848704*13^2 + O(13^20)q^{22} + (2468986169217895590399 + O(13^20))*a + 7607690842342525786757 + O(13^20)q^{23} + -(5259270433181216630409 + O(13^20))*a - 2191531923848513372433 + O(13^20)q^{24} + -(3220354663910245615171 + O(13^20))*a - 9224363017955273273802 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 2760368466423868014502 + O(13^20)x + 6971212808514269282042 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^10q^{2} + z^13q^{3} + z^11q^{4} + z^74q^{5} + z^23q^{6} + 6q^{7} + z^146q^{8} + z^3q^{9} + 12q^{10} + 11q^{11} + z^24q^{12} + z^80q^{14} + z^87q^{15} + z^68q^{16} + z^49q^{17} + z^13q^{18} + z^85q^{19} + z^85q^{20} + z^83q^{21} + z^108q^{22} + z^116q^{23} + z^159q^{24} + z^114q^{25} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^150q^{2} + z^69q^{3} + z^123q^{4} + z^158q^{5} + z^51q^{6} + 11q^{7} + z^62q^{8} + z^115q^{9} + 10q^{10} + 6q^{11} + z^24q^{12} + z^155q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^13)\omega^{1}&0\\0&u(z^155)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(9329958858288907968487 + O(13^20))*a + 7562562216112441623311 + O(13^20)q^{3} + -(2760368466423868014502 + O(13^20))*a - 6971212808514269314810 + O(13^20)q^{4} + (8267390115163250621502 + O(13^20))*a - 259606510531290228534 + O(13^20)q^{5} + -(207893217219449301505 + O(13^20))*a + 6557687179708102195828 + O(13^20)q^{6} + 823543q^{7} + -(7276293298716973815740 + O(13^20))*a - 6407957626558759662389 + O(13^20)q^{8} + -(4612632675558796866387 + O(13^20))*a + 3657550997986767747634 + O(13^20)q^{9} + -(127045723787799152879*13 + O(13^20))*a - 1313135677938326435025 + O(13^20)q^{10} + -(35477406620544946425*13^2 + O(13^20))*a + 511221811079119917931 + O(13^20)q^{11} + -(2870244219471936583651 + O(13^20))*a - 510752989406308633064*13 + O(13^20)q^{12} + (453054560525487782930*13 + O(13^20))*a - 636101102354146201010*13 + O(13^20)q^{13} + 823543*aq^{14} + (4045000997094555040251 + O(13^20))*a + 9091857072255346124734 + O(13^20)q^{15} + (33108491273964405228991 + O(13^21))*a - 104878616586830885839068 + O(13^21)q^{16} + (4002448281428558961755 + O(13^20))*a - 8052215476668073979351 + O(13^20)q^{17} + (4327911188296000936169 + O(13^20))*a + 3606810869568952698385 + O(13^20)q^{18} + -(2420627208219898283522 + O(13^20))*a + 4938669912157861915975 + O(13^20)q^{19} + -(7684293478524480068193 + O(13^20))*a + 1008598992274449635128 + O(13^20)q^{20} + (8526297385554817799655 + O(13^20))*a - 2498558523391503626036 + O(13^20)q^{21} + -(9288199545084343608467 + O(13^20))*a + 6185546384500848704*13^2 + O(13^20)q^{22} + (2468986169217895590399 + O(13^20))*a + 7607690842342525786757 + O(13^20)q^{23} + -(5259270433181216630409 + O(13^20))*a - 2191531923848513372433 + O(13^20)q^{24} + -(3220354663910245615171 + O(13^20))*a - 9224363017955273273802 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 2760368466423868014502 + O(13^20)x + 6971212808514269282042 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^1376q^{2} + z^810q^{3} + z^740q^{4} + z^1871q^{5} + z^2186q^{6} + 7q^{7} + z^1790q^{8} + z^292q^{9} + z^1051q^{10} + z^821q^{11} + z^1550q^{12} + z^1193q^{14} + z^485q^{15} + z^551q^{16} + z^974q^{17} + z^1668q^{18} + z^643q^{19} + z^415q^{20} + z^627q^{21} + zq^{22} + z^391q^{23} + z^404q^{24} + z^1066q^{25} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 11=0$.
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^1010q^{2} + z^1542q^{3} + z^8q^{4} + z^773q^{5} + z^356q^{6} + 2q^{7} + z^692q^{8} + z^1756q^{9} + z^1783q^{10} + z^455q^{11} + z^1550q^{12} + z^2006q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^190)\omega^{1}&0\\0&u(z^2006)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(9008477182565175429812 + O(13^20))*a^2 - (6930333744638621806994 + O(13^20))*a + 4300258995866419843421 + O(13^20)q^{3} + a^2 - 32768q^{4} + -(2964359928287402304331 + O(13^20))*a^2 - (1757721686401842603645 + O(13^20))*a - 4443798190065103286486 + O(13^20)q^{5} + (4998574995193868571287 + O(13^20))*a^2 - (7637852155287736050564 + O(13^20))*a + 9299819375120993796321 + O(13^20)q^{6} + -823543q^{7} + -(1906876503905461345339 + O(13^20))*a^2 - (6129181235575036689214 + O(13^20))*a + 6841595529072991540694 + O(13^20)q^{8} + (1793137601382775662480 + O(13^20))*a^2 - (40762911108046055139*13^2 + O(13^20))*a - 973902014274334146851 + O(13^20)q^{9} + -(5959567866928065320929 + O(13^20))*a^2 - (5057469441796534747190 + O(13^20))*a + 867459049674325195240 + O(13^20)q^{10} + (6351981275299786845756 + O(13^20))*a^2 - (2103830659479674766374 + O(13^20))*a + 2519601504799296911512 + O(13^20)q^{11} + (6954427907184748615104 + O(13^20))*a^2 + (873694955974738345564 + O(13^20))*a - 6886480903641220665519 + O(13^20)q^{12} + -(1030615802461359048306*13 + O(13^21))*a^2 - (79167870065788685880*13^2 + O(13^21))*a + 553311921083753685440*13^2 + O(13^21)q^{13} + -823543*aq^{14} + -(4664743818236403190836 + O(13^20))*a^2 - (7092915083513485855520 + O(13^20))*a + 7658356677567277172343 + O(13^20)q^{15} + -(37247579226513077920*13^2 + O(13^20))*a^2 + (135058337945574250392 + O(13^20))*a + 9104579632722349766733 + O(13^20)q^{16} + -(6143200634935647101434 + O(13^20))*a^2 - (571853563683940298298*13 + O(13^20))*a + 8207864993725851737170 + O(13^20)q^{17} + (4292732895182297414103 + O(13^20))*a^2 - (6144362799009061339864 + O(13^20))*a - 8258885600001175582669 + O(13^20)q^{18} + (4775097506603671335669 + O(13^20))*a^2 - (9002886334337849257354 + O(13^20))*a + 444957876281915954470*13 + O(13^20)q^{19} + -(5308630941184266793278 + O(13^20))*a^2 + (3871200439242191073082 + O(13^20))*a + 3373448483642680676881 + O(13^20)q^{20} + -(4359620071003933888449 + O(13^20))*a^2 + (9161898921877732053830 + O(13^20))*a - 6229530206187285965860 + O(13^20)q^{21} + -(1296639899694981309427 + O(13^20))*a^2 - (2108420887004687990167 + O(13^20))*a + 1404619961762949145807 + O(13^20)q^{22} + (1936349912447558024678 + O(13^20))*a^2 + (3824381964405959177977 + O(13^20))*a - 7344721669079266968810 + O(13^20)q^{23} + -(5706444204834258561613 + O(13^20))*a^2 + (4108826849791958911736 + O(13^20))*a + 2770118309626649129566 + O(13^20)q^{24} + -(6653555267272187151544 + O(13^20))*a^2 - (536848235660997762200 + O(13^20))*a - 493592215178014659534*13 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 1906876503905461345339 + O(13^20)x^{2} + 6129181235575036623678 + O(13^20)x + -6841595529072991540694 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^1964q^{2} + z^738q^{3} + z^2084q^{4} + z^2171q^{5} + z^506q^{6} + 7q^{7} + z^1658q^{8} + z^1036q^{9} + z^1939q^{10} + z^401q^{11} + z^626q^{12} + z^1781q^{14} + z^713q^{15} + z^887q^{16} + z^2102q^{17} + z^804q^{18} + z^1063q^{19} + z^2059q^{20} + z^555q^{21} + z^169q^{22} + z^199q^{23} + z^200q^{24} + z^82q^{25} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 11=0$.
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^1598q^{2} + z^1470q^{3} + z^1352q^{4} + z^1073q^{5} + z^872q^{6} + 2q^{7} + z^560q^{8} + z^304q^{9} + z^475q^{10} + z^35q^{11} + z^626q^{12} + z^830q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^1366)\omega^{1}&0\\0&u(z^830)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(9008477182565175429812 + O(13^20))*a^2 - (6930333744638621806994 + O(13^20))*a + 4300258995866419843421 + O(13^20)q^{3} + a^2 - 32768q^{4} + -(2964359928287402304331 + O(13^20))*a^2 - (1757721686401842603645 + O(13^20))*a - 4443798190065103286486 + O(13^20)q^{5} + (4998574995193868571287 + O(13^20))*a^2 - (7637852155287736050564 + O(13^20))*a + 9299819375120993796321 + O(13^20)q^{6} + -823543q^{7} + -(1906876503905461345339 + O(13^20))*a^2 - (6129181235575036689214 + O(13^20))*a + 6841595529072991540694 + O(13^20)q^{8} + (1793137601382775662480 + O(13^20))*a^2 - (40762911108046055139*13^2 + O(13^20))*a - 973902014274334146851 + O(13^20)q^{9} + -(5959567866928065320929 + O(13^20))*a^2 - (5057469441796534747190 + O(13^20))*a + 867459049674325195240 + O(13^20)q^{10} + (6351981275299786845756 + O(13^20))*a^2 - (2103830659479674766374 + O(13^20))*a + 2519601504799296911512 + O(13^20)q^{11} + (6954427907184748615104 + O(13^20))*a^2 + (873694955974738345564 + O(13^20))*a - 6886480903641220665519 + O(13^20)q^{12} + -(1030615802461359048306*13 + O(13^21))*a^2 - (79167870065788685880*13^2 + O(13^21))*a + 553311921083753685440*13^2 + O(13^21)q^{13} + -823543*aq^{14} + -(4664743818236403190836 + O(13^20))*a^2 - (7092915083513485855520 + O(13^20))*a + 7658356677567277172343 + O(13^20)q^{15} + -(37247579226513077920*13^2 + O(13^20))*a^2 + (135058337945574250392 + O(13^20))*a + 9104579632722349766733 + O(13^20)q^{16} + -(6143200634935647101434 + O(13^20))*a^2 - (571853563683940298298*13 + O(13^20))*a + 8207864993725851737170 + O(13^20)q^{17} + (4292732895182297414103 + O(13^20))*a^2 - (6144362799009061339864 + O(13^20))*a - 8258885600001175582669 + O(13^20)q^{18} + (4775097506603671335669 + O(13^20))*a^2 - (9002886334337849257354 + O(13^20))*a + 444957876281915954470*13 + O(13^20)q^{19} + -(5308630941184266793278 + O(13^20))*a^2 + (3871200439242191073082 + O(13^20))*a + 3373448483642680676881 + O(13^20)q^{20} + -(4359620071003933888449 + O(13^20))*a^2 + (9161898921877732053830 + O(13^20))*a - 6229530206187285965860 + O(13^20)q^{21} + -(1296639899694981309427 + O(13^20))*a^2 - (2108420887004687990167 + O(13^20))*a + 1404619961762949145807 + O(13^20)q^{22} + (1936349912447558024678 + O(13^20))*a^2 + (3824381964405959177977 + O(13^20))*a - 7344721669079266968810 + O(13^20)q^{23} + -(5706444204834258561613 + O(13^20))*a^2 + (4108826849791958911736 + O(13^20))*a + 2770118309626649129566 + O(13^20)q^{24} + -(6653555267272187151544 + O(13^20))*a^2 - (536848235660997762200 + O(13^20))*a - 493592215178014659534*13 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 1906876503905461345339 + O(13^20)x^{2} + 6129181235575036623678 + O(13^20)x + -6841595529072991540694 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^320q^{2} + z^1746q^{3} + z^836q^{4} + z^167q^{5} + z^2066q^{6} + 7q^{7} + z^1310q^{8} + z^1600q^{9} + z^487q^{10} + z^1889q^{11} + z^386q^{12} + z^137q^{14} + z^1913q^{15} + z^575q^{16} + z^1682q^{17} + z^1920q^{18} + z^1771q^{19} + z^1003q^{20} + z^1563q^{21} + z^13q^{22} + z^691q^{23} + z^860q^{24} + z^682q^{25} + \cdots \in S_{16}(\Gamma_0(7);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{3} + 2x + 11=0$.
We have $\rho_f \otimes \chi^{10} \sim \rho_g$, where $g$ is the
weight-$12$ newform
$g = q + z^2150q^{2} + z^282q^{3} + z^104q^{4} + z^1265q^{5} + z^236q^{6} + 2q^{7} + z^212q^{8} + z^868q^{9} + z^1219q^{10} + z^1523q^{11} + z^386q^{12} + z^1922q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^274)\omega^{1}&0\\0&u(z^1922)\omega^{2}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + -(9008477182565175429812 + O(13^20))*a^2 - (6930333744638621806994 + O(13^20))*a + 4300258995866419843421 + O(13^20)q^{3} + a^2 - 32768q^{4} + -(2964359928287402304331 + O(13^20))*a^2 - (1757721686401842603645 + O(13^20))*a - 4443798190065103286486 + O(13^20)q^{5} + (4998574995193868571287 + O(13^20))*a^2 - (7637852155287736050564 + O(13^20))*a + 9299819375120993796321 + O(13^20)q^{6} + -823543q^{7} + -(1906876503905461345339 + O(13^20))*a^2 - (6129181235575036689214 + O(13^20))*a + 6841595529072991540694 + O(13^20)q^{8} + (1793137601382775662480 + O(13^20))*a^2 - (40762911108046055139*13^2 + O(13^20))*a - 973902014274334146851 + O(13^20)q^{9} + -(5959567866928065320929 + O(13^20))*a^2 - (5057469441796534747190 + O(13^20))*a + 867459049674325195240 + O(13^20)q^{10} + (6351981275299786845756 + O(13^20))*a^2 - (2103830659479674766374 + O(13^20))*a + 2519601504799296911512 + O(13^20)q^{11} + (6954427907184748615104 + O(13^20))*a^2 + (873694955974738345564 + O(13^20))*a - 6886480903641220665519 + O(13^20)q^{12} + -(1030615802461359048306*13 + O(13^21))*a^2 - (79167870065788685880*13^2 + O(13^21))*a + 553311921083753685440*13^2 + O(13^21)q^{13} + -823543*aq^{14} + -(4664743818236403190836 + O(13^20))*a^2 - (7092915083513485855520 + O(13^20))*a + 7658356677567277172343 + O(13^20)q^{15} + -(37247579226513077920*13^2 + O(13^20))*a^2 + (135058337945574250392 + O(13^20))*a + 9104579632722349766733 + O(13^20)q^{16} + -(6143200634935647101434 + O(13^20))*a^2 - (571853563683940298298*13 + O(13^20))*a + 8207864993725851737170 + O(13^20)q^{17} + (4292732895182297414103 + O(13^20))*a^2 - (6144362799009061339864 + O(13^20))*a - 8258885600001175582669 + O(13^20)q^{18} + (4775097506603671335669 + O(13^20))*a^2 - (9002886334337849257354 + O(13^20))*a + 444957876281915954470*13 + O(13^20)q^{19} + -(5308630941184266793278 + O(13^20))*a^2 + (3871200439242191073082 + O(13^20))*a + 3373448483642680676881 + O(13^20)q^{20} + -(4359620071003933888449 + O(13^20))*a^2 + (9161898921877732053830 + O(13^20))*a - 6229530206187285965860 + O(13^20)q^{21} + -(1296639899694981309427 + O(13^20))*a^2 - (2108420887004687990167 + O(13^20))*a + 1404619961762949145807 + O(13^20)q^{22} + (1936349912447558024678 + O(13^20))*a^2 + (3824381964405959177977 + O(13^20))*a - 7344721669079266968810 + O(13^20)q^{23} + -(5706444204834258561613 + O(13^20))*a^2 + (4108826849791958911736 + O(13^20))*a + 2770118309626649129566 + O(13^20)q^{24} + -(6653555267272187151544 + O(13^20))*a^2 - (536848235660997762200 + O(13^20))*a - 493592215178014659534*13 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{3} + 1906876503905461345339 + O(13^20)x^{2} + 6129181235575036623678 + O(13^20)x + -6841595529072991540694 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^3q^{2} + z^27q^{3} + z^47q^{4} + z^95q^{5} + z^30q^{6} + 10q^{7} + z^131q^{8} + z^113q^{9} + 11q^{10} + z^111q^{11} + z^74q^{12} + z^143q^{14} + z^122q^{15} + z^83q^{16} + z^104q^{17} + z^116q^{18} + z^83q^{19} + z^142q^{20} + z^167q^{21} + z^114q^{22} + z^35q^{23} + z^158q^{24} + z^83q^{25} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
$g = q + z^115q^{2} + z^139q^{3} + z^103q^{4} + z^95q^{5} + z^86q^{6} + 4q^{7} + z^131q^{8} + zq^{9} + 8q^{10} + z^55q^{11} + z^74q^{12} + z^118q^{13} + \cdots .$
We deduce that the semisimplification of $\rho_f|D_\ell$ is
$$\left(\begin{matrix}u(z^50)\omega^{1}&0\\0&u(z^118)\omega^{4}\end{matrix}\right),$$
where $u$ is unramified and $\omega$ is the mod-$\ell$ cyclotomic character. The form
$f_0 = q + aq^{2} + (4135691119088469399136 + O(13^20))*a + 5615421483611304172489 + O(13^20)q^{3} + -(8953618332239728730471 + O(13^20))*a + 167263672957981951779 + O(13^20)q^{4} + (97040373503270447720454 + O(13^21))*a + 121187770553216030323515 + O(13^21)q^{5} + -(23126313581007614138893 + O(13^21))*a - 28561551487026734918472 + O(13^21)q^{6} + -5764801q^{7} + (4090575135417907098201 + O(13^20))*a - 3256671757963281841429 + O(13^20)q^{8} + (1203441815515054614581 + O(13^20))*a + 894543179245889120553 + O(13^20)q^{9} + (12103911822807321977*13 + O(13^20))*a - 7068627447387227516043 + O(13^20)q^{10} + (3182180005789565585868 + O(13^20))*a - 7650712860188665902082 + O(13^20)q^{11} + -(7799936456098822206647 + O(13^20))*a - 4048875828633923707073 + O(13^20)q^{12} + -(334823407601103396372*13 + O(13^20))*a - 425389944685188984257*13 + O(13^20)q^{13} + -5764801*aq^{14} + (8296034556876916673741 + O(13^20))*a - 2940860462411217097534 + O(13^20)q^{15} + -(5038973355310959427058 + O(13^20))*a - 7857079557472973136681 + O(13^20)q^{16} + (4676889142546149647297 + O(13^20))*a + 4948958855141927397983 + O(13^20)q^{17} + -(5434069693852536120994 + O(13^20))*a + 3297700911199274468922 + O(13^20)q^{18} + -(623651140214688620548 + O(13^20))*a - 3891405350598755450911 + O(13^20)q^{19} + -(4107806623478057653943 + O(13^20))*a + 8836909102253628070102 + O(13^20)q^{20} + (5682119012203375660549 + O(13^20))*a - 5422584893154377535956 + O(13^20)q^{21} + -(111645267936654784147045 + O(13^21))*a - 56360267065638431812394 + O(13^21)q^{22} + -(7443594906951534726176 + O(13^20))*a + 5676537174290327554516 + O(13^20)q^{23} + (7933365861727221063351 + O(13^20))*a + 4419559733912660929908 + O(13^20)q^{24} + -(9441201803695877715325 + O(13^20))*a + 9004167038026633551772 + O(13^20)q^{25} + \cdots$
is an $\ell$-adic newform that lifts~$f$.
Here the coefficients of $f_0$ are written in terms of a root~$a$ of $x^{2} + 8953618332239728730471 + O(13^20)x + -167263672957982082851 + O(13^20)$.
The slope of $f_0$ is $1$.

\item  Consider
$f = q + z^39q^{2} + z^15q^{3} + z^107q^{4} + z^59q^{5} + z^54q^{6} + 10q^{7} + z^23q^{8} + z^125q^{9} + 11q^{10} + z^99q^{11} + z^122q^{12} + z^11q^{14} + z^74q^{15} + z^71q^{16} + z^8q^{17} + z^164q^{18} + z^71q^{19} + z^166q^{20} + z^155q^{21} + z^138q^{22} + z^119q^{23} + z^38q^{24} + z^71q^{25} + \cdots \in S_{18}(\Gamma_0(7);\overline{\mathbf{F}}_{13}).$
Also,~$z$ satisfies the equation $x^{2} + 12x + 2=0$.
We have $\rho_f \otimes \chi^{8} \sim \rho_g$, where $g$ is the
weight-$10$ newform
\$ g = q + z^151q^{2} + z^127q^{3} + z^163q^{4} + z^59q^{5} + z^110q^{6} + 4q^{7} + z^23q^{8} + z^13q^{9} + 8q^{10} + z^43q^{11} + z^122q^{12} + z^22q^{13} +  \cdot