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% nonmaximal.tex  ((c) William A. Stein, 1999)
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\myhead{The first few nonmaximal orders attached\vspace{.3ex}\\
to weight two newforms on $\Gamma_0(N)$}{0.1}{}

\section*{Introduction}
Let $f \in S_2(\Gamma_0(N))$ be a weight two newform.
Attach to $f$ the ring
$$\O_f = \Z[\ldots a_n \ldots]$$
generated by the Fourier
coefficients of $f$.  This ring is an order
in the totally real number field $K_f=\Q[\ldots a_n \ldots]$.
David Carleton asked me for examples in which $\O_f$
is not equal to the maximal order in $K_f$, equivalently,
for which $\Spec(\O_f)$ is not normal.
Armand Brumer pointed out that Birch had long
ago laboriously compiled tables of $f$ so that
$\O_f$ appeared to be non-maximal.
it was perhaps possible that in  Birch's examples,
given the limited computing resources of the day, that
he had not computed enough of the $a_n$'s to generate
the full ring $\O_f$.  When $N=1$ and the weight
grows, Jochnowitz showed that $\O_f$ is far from
normal.

\section*{Data gathering}
Using \hecke{} \cite{stein:hecke}, the author computed the index
of $\O_f$ in the maximal order for every newform $f$
of level $N\leq 450$.   This was done as follows:
\begin{enumerate}
\item Compute enough $a_n$ to generate $\O_f$ as a $\Z$-module
using the Sturm bound
\cite{generatinghecke, stein:congruence,  sturm:cong}.
\item Compute the discriminant of $\O_f$ by embedding
a matrix representation of $\O_f$ into a space of column vectors,
finding an integral basis,  and computing the
determinant of the trace pairing.
\item Compute the discriminant of $K_f$ using \lidia{}.
\end{enumerate}

There are $1775$ newforms of level $N\leq 450$.
Of these, $93$ have the property that $\O_f$ is {\em not} maximal.
The distribution is as follows, where index is the index of
$\O_f$ in the maximal order.
\begin{center}
{\bf Table 1. Distribution of indexes\vspace{1.5ex}}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
index & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16\\\hline
number &1682&77&1&12&1&0&0&1&0&0&0&0&0&0&0&1\\
\hline\end{tabular}
\end{center}

\vspace{2ex}

\begin{center}
{\bf Table 2. Each newform of level $\leq 450$ with
$\O_f$ having given index:\vspace{1.5ex}}
\begin{tabular}{|c|l|}\hline
index & $f$\\\hline\hline
2 &69B, 77D, 105B, 136C, 138D, 154D, 160C, 165C, 171E, 195E, 207D, 213E, \\
& 217D, 221D, 221G, 224C, 224D, 225F, 226D, 238F, 255D, 256E, 260B, 272E,\\
& 273E, 282E, 299B, 299G, 301D, 305C, 305D, 310E, 315F, 320G, 322E, 322G,\\
& 323E, 329F, 337B, 340B, 355E, 357H, 359D, 363G, 368I, 376D, 377F, 385H,\\
& 390H, 392G, 399F, 399G, 406G, 410E, 410G, 410I, 414F, 415D, 416D, 417F,\\
& 426G, 429H, 433C, 434I, 435E, 435J, 437F, 438H, 438I, 442F, 442H, 442I,\\
& 445A, 445F, 445G, 448I, 448J\\\hline
3 &271B\\\hline
4 &219E, 291H, 293B, 303E, 371E, 387J, 389E, 395H, 413F, 416F, 431F, 437H\\\hline
5 &  401B\\\hline
8 & 371F\\\hline
16 & 257B\\
\hline\end{tabular}
\end{center}

\vspace{2ex}

\begin{center}
{\bf Table 3. The first example exhibiting each index.}
$$\begin{array}{|c|l|c|c|}\hline \text{index} & f & [K_f:\Q] & \disc(K_f) \\\hline\hline 2 & \text{69B} & 2 & 5 \\ 3 & \text{271B} & 16 & 1367\cdot6091\cdot1132673\cdot14171513\cdot 172450541\\ 4 & \text{219E} & 6 & 2^2\cdot 1189637 \\ 5 & \text{401B} & 21 & 2^8\cdot19\cdot 163\cdot71742740351\cdot 388881803749\cdot 34393898968391\\ 8 & \text{371F} & 11 & 2^8\cdot 157\cdot 76723322773093\\ 16 & \text{257B} & 14 & 2^7\cdot 29\cdot479\cdot71711\cdot 409177\cdot654233\\ \hline\end{array}$$
\end{center}

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