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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. 
Unfortunately, we do not have access to Birch's tables and
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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