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Author: William A. Stein
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\documentclass{article}
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\include{macros}
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\title{Approximating twists by finite slope newforms}
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\author{Robert Coleman\footnote{Coleman hasn't read this version yet,
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so any mistakes are Stein's.} \and William A.~Stein}
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\begin{document}
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\maketitle
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\section{Introduction}
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We have experimentally investigated the question of approximation
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of twists of finite slope forms.
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Let~$p$ be a prime and let~$N$ be a positive
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integer not divisible by~$p$.
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Consider a classical newform~$f$ in
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$S_k(\Gamma_0(N),\Zp) := S_k(\Gamma_0(N),\Z)\tensor\Zp.$
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The \defn{slope} of~$f$ is $\ord_p(a_p(f))$.
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By~\cite[Prop.~3.6]{shimura:intro}, the twist of~$f$ by a Dirichlet character
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$\eps:(\Z/p\Z)^*\ra \C^*$
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is an eigenform on $\Gamma_1(Np^{2r})$; the slope of this
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twist is infinite.
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Assume that~$f$ has finite slope.
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Is it possible to approximate $f^{\eps}$ by forms of finite
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slope? One way to make approximate'' precise and falsifiable,
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is to ask the following: let $n\geq 1$ be an integer;
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does there exist a newform
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$g \in S_{k + \vphi(p^n)}(\Gamma_0(Np);\Z_p)$
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such that $g \con f^{\eps}\pmod{p^n}$?
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(By this congruence, we mean that
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$\ord_p(g -f^\eps) = \min_{i\geq 1}\{\ord_p(a_i(g)-a_i(f^\eps))\} \geq n.$)
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The question of approximation of twists by finite slope forms
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is motivated by our attempt to complete the
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eigencurve~\cite{eigencurve}, and by questions
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of N.~Jochnowitz.
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If the above question has an affirmative
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answer for $r=1$, then we obtain a sequence $\{g_n\}$ of
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finite slope classical overconvergent forms that converges
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to the infinite slope form $f^{\eps}$.
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An unusual feature of the \nobreak{computations} described in this
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paper is that we approximate~$f$ by {\em newforms} on $\Gamma_0(Np)$, so the
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slopes increase quickly.
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\section{A false conjecture}
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In this section we state a very general, {\em and very false}, conjecture.
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Let~$p$ be a prime and let~$N$ be a positive
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integer not divisible by~$p$.
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\begin{conjecture}[False conjecture]\label{conj:false}
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Let $f \in S_k(\Gamma_0(N),\Zp)$ be a finite slope newform,
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and~$\eps$ a character mod~$p$ such that $\eps^2=1$.
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Then for all $n\geq 1$ there
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exists an eigenform
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$g \in S_{k + \vphi(p^n)}(\Gamma_0(Np);\Z_p)$
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such that $g \con f^{\eps}\pmod{p^n}$.
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\end{conjecture}
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\section{Numerical experiments}
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\label{sec:plan}
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In the remainder of this paper we report on numerical experiments
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designed to gather evidence about Conjecture~\ref{conj:false}.
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These computations were carried out in \magma{} (cf.~\cite{magma})
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using algorithms described in~\cite{stein:phd}.
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We perform the following experiments:
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\begin{dashlist}
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\item Section~\ref{exp:delta}.
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Let $f=\Delta$ be the unique cusp form of weight~$12$.
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For each $p=2,3,5,7,11$ consider the twist $f^{\eps}$ by either the
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trivial or quadratic mod~$p$ character.
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We compute $\ord_p(g- f^{\eps})$ for each rational
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newform $g\in S_{k+\vphi(p^n)}(\Gamma_0(Np);\Z_p)$,
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for the first few values of~$n$.
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In fact, $\ord_p(g-f^{\eps})$ was computed using only~$17$ terms of the
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$q$-expansion, which is probably enough, but we have {\em not}
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proved this.
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\item Section~\ref{exp:11}.
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Let~$f$ be the weight-$2$ newform
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corresponding to the elliptic curve $X_0(11)$.
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For $p=2,3,5$ we perform the same computation as for~$\Delta$.
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\end{dashlist}
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\vspace{1ex}
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Since, in this experiment, we only twist $f$ by the trivial
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or quadratic character, we adopt the notation $f^{(1)}$ and
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$f^{(-1)}$ for these twists, respectively.
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Thus $f^{(1)}$ is the $p$-deprivation
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of~$f$, and $f^{(-1)}$ is the quadratic twist.
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\begin{remark}
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The newforms at level~$Np$ all have slope $(k-2)/2$
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(cf.~\cite[\S6]{diamond-im}).
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\end{remark}
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\begin{remark}
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Some of the tables are missing entries; these are being computed...
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\end{remark}
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\subsection{Twisting~$\Delta$}\label{exp:delta}
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Consider the weight-12 cuspform $\Delta$; the $q$-expansion of $\Delta$ is
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\begin{eqnarray*}
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\Delta&=&
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\prod(1-q^n)^{24} \\
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&=&q - 24q^2 + 252q^3 - 1472q^4 + 4830q^5 - 6048q^6 - 16744q^7 +\cdots.
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\end{eqnarray*}
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Let~$p$ be a prime and let $\eps:(\Z/p\Z)^*\ra\{\pm 1\}$ be a character;
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then $\eps^2=1$, so $\Delta^{\eps}\in S_{12}(\Gamma_0(9))$.
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As outlined in Section~\ref{sec:plan}, we compute each of the
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rational newforms $g \in S_{12+\vphi(p^n)}(\Gamma_0(p),\Z_p),$
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and then $\ord_p(\Delta^{\eps}-g)$.
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The computations are given in the following tables.
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For example, the 4th line of the first table below
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says that when $n=4$ there are two
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$\Z_2$-rational newforms
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$g_1$ and $g_2$; and that $\Delta^{(1)} \con g_1 \pmod{2^5}$
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and $\Delta^{(1)} \con g_2 \pmod{2^5}$.
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The non-$\Z_p$-rational newforms in $S_{12+\vphi(p^n)}(\Gamma_0(p),\Zbar_p)$
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are not listed.
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(2)$ congruent mod $2$ to the
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$2$-deprivation of $\Delta$.\vspace{1ex}\\
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\begin{tabular}{l|l}
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$n\quad$ & $\quad\ord_2(\Delta^{(1)}-g_i)$\\\hline
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$1$ & none\\
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$2$ & 3, 3\\
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$3$ & 4\\
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$4$ & 5, 5\\
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$5$ & 6, 6\\
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$6$ & 7, 7, 7, 7
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\end{tabular}
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\end{center}
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\newpage
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(3)$ congruent mod $3$ to
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twists of $\Delta$\vspace{1ex}\\
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\begin{tabular}{l|l|l}
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$n\quad$ & $\quad\ord_3(\Delta^{(1)}-g_i)$ & $\quad\ord_3(\Delta^{(-1)}-g_i)$
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\\\hline
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$1$ & 1, 1, 1 & 1, 1, 1\\
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$2$ & 1, 2, 1 & 2, 1, 2 \\
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$3$ & 3, 1, 3, 3, 1 &1, 3, 1, 1, 3\\
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$4$ & 4, 4, 1, 1, 1, 3, 4, 4, 1, 1, 1
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& 1, 1, 4, 4, 3, 1, 1, 1, 4, 4, 3
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\end{tabular}
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\end{center}
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(5)$ congruent mod $5$ to
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twists of $\Delta$\vspace{1ex}\\
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\begin{tabular}{l|l|l}
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$n\quad$ & $\quad\ord_5(\Delta^{(1)}-g_i)$ & $\quad\ord_5(\Delta^{(-1)}-g_i)$
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\\\hline
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$1$ & 1, 0, 1, 0, 0 & 0, 1, 0, 1, 1\\
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$2$ & 2, 1, 1, 0, 0, 2, 1, 1, 0, 0, 0& 0, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2\\
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\end{tabular}
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\end{center}
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(7)$ congruent mod $7$ to twists of $\Delta$
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\vspace{1ex}\\
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\begin{tabular}{l|l}
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$n\quad$ & $\quad\ord_7(\Delta^{(1)}-g_i)$, $\quad\ord_7(\Delta^{(-1)}-g_i)$
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\\\hline
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$1$ & [1, 1, 0, 0, 1, 1, 0, 0, 0], [1, 1, 0, 0, 1, 1, 0, 0, 0] \\
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$2$ & [1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], \\
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& [1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0]
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\end{tabular}
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\end{center}
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(11)$ congruent mod $11$ to twists of $\Delta$
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\vspace{1ex}\\
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\begin{tabular}{l|l}
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$n\quad$ & $\quad\ord_{11}(\Delta^{(1)}-g_i)$,
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$\quad\ord_{11}(\Delta^{(-1)}-g_i)$
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\\\hline
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$1$ & [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],\\
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& [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ] \\
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$2$ & \\
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\end{tabular}
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\end{center}
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\comment{
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\begin{remark}
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There seems to be a form~$F$ of weight~$30$ congruent to $\Delta \pmod{27}$
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and a form~$G$ congruent to $\Delta$ $3$-deprived, $\Delta(3)$,
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modulo~$27$ of slope~$14$. But the evil twin
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of~$F$ is also congruent to $\Delta(3) \pmod{27}$
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and it has slope~$27$! The evil twin is an old eigenform in
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the space $S_{30}(\Gamma_0(3),\Z_3)$.
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\end{remark}}
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\subsection{Twisting the weight-$2$ newform on $X_0(11)$}\label{exp:11}
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Let~$f$ be the form corresponding to the elliptic curve of conductor~$11$.
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Fix a prime~$p$, and let $\eps$ be the trivial character modulo~$p$.
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We list the congruences between $f^{\eps}$ and the $\Z_2$-rational
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newforms in $S_{2+\vphi(p^n)}(\Gamma_0(p\cdot 11))$.
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For example, the 2nd line of the table below
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says that when $n=2$ there are three
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$\Z_2$-rational newforms
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$g_1,g_2,g_3$; and that $f^{(1)} \con g_1 \pmod{3}$,
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$f^{(1)} \not\con g_2 \pmod{3}$,
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and $f^{(1)} \con g_3 \pmod{3}$.
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The 3rd line says there are five $\Z_2$-rational newforms
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in $S_{2+\vphi(2^3)}(\Gamma_0(22))$,
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and the first, $g$, satisfies $f^{(1)}\con g\pmod{3^2}$.
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(22)$ congruent mod~$2$ to
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the $2$-deprivation of~$f$\vspace{1ex}\\
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\begin{tabular}{l|l}
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$n\quad$ & $\quad\ord_2(f^{(1)}-g_i)$\\\hline
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$1$ & none\\
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$2$ & 1, 0, 1\\
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$3$ & 2, 1, 0, 1, 2 \\
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$4$ & 1, 1, 3, 3, 0 \\
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$5$ & 4, 1, 0, 4, 0, 1, 0
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\end{tabular}
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\end{center}
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\newpage
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(33)$ congruent mod $3$ to the
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$3$-deprivation of~$f$\vspace{1ex}\\
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\begin{tabular}{l|l}
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$n\quad$ & $\quad\ord_3(f^{(1)}-g_i)$\\\hline
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$1$ & 0, 1, 1, 0 \\
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$2$ & 1, 0, 0, 1, 0, 0 \\
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$3$ & 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0
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\end{tabular}
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\end{center}
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(33)$ congruent mod $3$ to the
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quadratic twist of~$f$\vspace{1ex}\\
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\begin{tabular}{l|l}
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$n\quad$ & $\quad\ord_3(f^{(-1)}-g_i)$\\\hline
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$1$ & 1, 0, 0, 0\\
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$2$ & 0, 2, 1, 0, 1, 0\\
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$3$ & 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1
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\end{tabular}
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\end{center}
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(55)$ congruent mod $5$
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to twists of~$f$\vspace{1ex}\\
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\begin{tabular}{l|l|l}
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$n\quad$ & $\quad\ord_5(f^{(-1)}-g_i)$ & $\quad\ord_5(f^{(-1)}-g_i)$ \\\hline
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$1$ & 1, 0 & 0, 0\\
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$2$ & 0, 2, 1, 0, 1, 0 & \\
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\end{tabular}
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\end{center}
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\subsection{Twisting $X_0(14)$}\label{exp:14}
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\begin{center}
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\vspace{1ex}
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Eigenforms on $\Gamma_0(3\cdot 14)$ congruent mod $3$
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to twists of $h\in S_2(\Gamma_0(14))$\vspace{1ex}\\
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\begin{tabular}{l|l|l}
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$n\quad$ & $\quad\ord_3(h^{(1)}-g_i)$& $\quad\ord_3(h^{(-1)}-g_i)$
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\\\hline
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$1$ & 0, 1 & 0, 0\\
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$2$ & 0, 1, 0, 0, 0, 0 & 0, 0, 2, 0, 2, 0
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\end{tabular}
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\end{center}
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\bibliographystyle{amsalpha}
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\bibliography{biblio}
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\end{document}
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Dear Robert,
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I did a few more computations, up to weight 12+phi(3^4),
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and things look perfect. Can you prove Conjecture 1.1 in
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the attached file? Shall we plan a big systematic computation
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to test a twisting conjecture''?
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Best,
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William
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5 Oct 1999:
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Dear Robert,
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Yes, this is correct.
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Maybe we can fortmulate a conjecture for infinite slope forms which would
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implie the GM conjecturer but is not implied by it. Here is a
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false conjecture
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\proclaim False Conjecture. Let F(q) be the q-expansion of a weight k
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tame level N infinitre slope overconvergent p-adic eigenform. Then
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for intgers r and n\ge 1 (r,p)=1, there exists an overconvergent
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eigenform of weight k+r\phi(p^n) of slope n and tamre level N such that
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G(q) is congruent to F(q) modulo p^n.
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Remark. We know if n < the weight of G minus one, G must be
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classical.
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GM says: if d(k,\alp)=1 ("obvious" notation) and F is as form of weight
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k and slope \alp, then for every integer of k'\ge k congruent to k modulo
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\phi(p^n), n\ge\alp , there exists G of weight k' and
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slope \alp such that
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G(q)\con F(q) modulo p^n
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When p=3 and F=\Delta, this implies there exists a form of weight 2*9
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+12 of slope 2 congruernt to \Delta modulo 27.
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I just checked that this is true.
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You found one of slope \ge 3 congruent to the twist of
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Delta by \omega of this weight.
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What was the slope of the form you found?
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Over Z_3 there are five distinct eigenforms at weight 30.
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Two of them, g and h, are congruent modulo 27 to the twist of
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Delta by omega. The slope of g is 14 (i.e., ord_3(a_3(g))=14.),
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and the slope of h is also FOURTEEN.
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This is interesting: let's see what the slopes are for the
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other examples:
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slopes
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--------
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k = 12: 5
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k = 14: 6
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k = 30: 14, 14
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k = 66: 32,32, 32,32
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In general, by Atkin-Lehner theory, because we are considering
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newforms on Gamma_0(3), the slope is (k-2)/2. [See e.g.,
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page 64 of Diamond-Im].
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Is there a form of this weight congruent
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to the twist of \Delta by \omega^2, i.e., thre
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3-deprived version of \Delta, \Delta(3)?
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Let depriv := Delta x omega^2.
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First, at weight 12 there is a form congruent to deprive mod 3, but not
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modulo 3^2.
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Next, weight 14: There is a form congruent mod 3, but NOT modulo 3^2.
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Next, weight 30: There are three forms cong. mod 3^3 to deprive.
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Next, weight 66: There are forms cong. mod 3^4 to deprive.
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Is there a form of weight 6 congruent to \Delta(3) modulo 9?
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YES, there is a weight 6 form congruent to \Delta(3) modulo 9, but not
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modulo 27.
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We might be heading to another onjectural generalization of
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Hida theory.
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Great!
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William
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Buffers Files Tools Edit Search Mule Help
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There seems to be a form F of weight 30 congruent to \Delta mod 27
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and a form G congruent to \Delta 3-deprived, \Delta(3), mod 27 of
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slope 14. But the evil twin of F is also congruent to \Delta(3) mod
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27 and it has slope 27!
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Amazing!
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Let f be the form corresponding to the ellptic curve of conductor 11.
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Let f(2) be the 2-deprivation of f. I wrote a program to list the
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Z_2-rational newforms in S_{2+Phi(2^n)}(Gamma_0(2*11)). The third
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line of the following table says that when n=3 there are 3 Z_2 rational
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newforms h1,h2,h3; and that f(2) = h1 (mod 3), f(2) =/= h2 (mod 3),
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and f(2) = h3 (mod 3).
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p = 2, f in S_2(Gamma_0(11)).
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n | congruences
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-----------------------------
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0 | none
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1 | none
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2 | [ 1, 0, 1 ]
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3 | [ 2, 1, 0, 1, 2 ]
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4 | [ 1, 1, 3, 3, 0 ]
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5 | [ 4, 1, 0, 4, 0, 1, 0 ]
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Amazingly there are forms in S_{2+Phi(2^5)}(Gamma_0(22)) that
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are congruent mod 2^4 to f(2)!
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Note: Because I am only considering newforms, the slopes are (k-2)/2.
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p = 3, f in S_2(Gamma_0(11)).
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n | congruences
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-----------------------------
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0 | none
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1 | [ 0, 1, 1, 0 ]
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2 | [ 1, 0, 0, 1, 0, 0 ]
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3 |
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William
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