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% nonmaximal.tex ((c) William A. Stein, 1999)
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\begin{document}
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\myhead{The first few nonmaximal orders attached\vspace{.3ex}\\
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to weight two newforms on $\Gamma_0(N)$}{0.1}{}
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\section*{Introduction}
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Let $f \in S_2(\Gamma_0(N))$ be a weight two newform.
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Attach to $f$ the ring
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$$\O_f = \Z[\ldots a_n \ldots]$$
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generated by the Fourier
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coefficients of $f$. This ring is an order
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in the totally real number field $K_f=\Q[\ldots a_n \ldots]$.
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David Carleton asked me for examples in which $\O_f$
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is not equal to the maximal order in $K_f$, equivalently,
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for which $\Spec(\O_f)$ is not normal.
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Armand Brumer pointed out that Birch had long
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ago laboriously compiled tables of $f$ so that
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$\O_f$ appeared to be non-maximal.
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Unfortunately, we do not have access to Birch's tables and
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it was perhaps possible that in Birch's examples,
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given the limited computing resources of the day, that
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he had not computed enough of the $a_n$'s to generate
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the full ring $\O_f$. When $N=1$ and the weight
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grows, Jochnowitz showed that $\O_f$ is far from
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normal.
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\section*{Data gathering}
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Using \hecke{} \cite{stein:hecke}, the author computed the index
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of $\O_f$ in the maximal order for every newform $f$
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of level $N\leq 450$.` This was done as follows:
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\begin{enumerate}
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\item Compute enough $a_n$ to generate $\O_f$ as a $\Z$-module
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using the Sturm bound
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\cite{generatinghecke, stein:congruence, sturm:cong}.
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\item Compute the discriminant of $\O_f$ by embedding
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a matrix representation of $\O_f$ into a space of column vectors,
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finding an integral basis, and computing the
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determinant of the trace pairing.
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\item Compute the discriminant of $K_f$ using \lidia{}.
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\end{enumerate}
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There are $1775$ newforms of level $N\leq 450$.
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Of these, $93$ have the property that $\O_f$ is {\em not} maximal.
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The distribution is as follows, where index is the index of
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$\O_f$ in the maximal order.
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\begin{center}
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{\bf Table 1. Distribution of indexes\vspace{1.5ex}}
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\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
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index & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16\\\hline
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number &1682&77&1&12&1&0&0&1&0&0&0&0&0&0&0&1\\
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\hline\end{tabular}
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\end{center}
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\vspace{2ex}
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\begin{center}
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{\bf Table 2. Each newform of level $\leq 450$ with
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$\O_f$ having given index:\vspace{1.5ex}}
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\begin{tabular}{|c|l|}\hline
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index & $f$\\\hline\hline
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2 &69B, 77D, 105B, 136C, 138D, 154D, 160C, 165C, 171E, 195E, 207D, 213E, \\
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& 217D, 221D, 221G, 224C, 224D, 225F, 226D, 238F, 255D, 256E, 260B, 272E,\\
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& 273E, 282E, 299B, 299G, 301D, 305C, 305D, 310E, 315F, 320G, 322E, 322G,\\
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& 323E, 329F, 337B, 340B, 355E, 357H, 359D, 363G, 368I, 376D, 377F, 385H,\\
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& 390H, 392G, 399F, 399G, 406G, 410E, 410G, 410I, 414F, 415D, 416D, 417F,\\
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& 426G, 429H, 433C, 434I, 435E, 435J, 437F, 438H, 438I, 442F, 442H, 442I,\\
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& 445A, 445F, 445G, 448I, 448J\\\hline
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3 &271B\\\hline
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4 &219E, 291H, 293B, 303E, 371E, 387J, 389E, 395H, 413F, 416F, 431F, 437H\\\hline
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5 & 401B\\\hline
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8 & 371F\\\hline
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16 & 257B\\
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\hline\end{tabular}
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\end{center}
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\vspace{2ex}
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\begin{center}
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{\bf Table 3. The first example exhibiting each index.}
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$$\begin{array}{|c|l|c|c|}\hline
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\text{index} & f & [K_f:\Q] & \disc(K_f) \\\hline\hline
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2 & \text{69B} & 2 & 5 \\
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3 & \text{271B} & 16 & 1367\cdot6091\cdot1132673\cdot14171513\cdot 172450541\\
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4 & \text{219E} & 6 & 2^2\cdot 1189637 \\
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5 & \text{401B} & 21 & 2^8\cdot19\cdot 163\cdot71742740351\cdot
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388881803749\cdot 34393898968391\\
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8 & \text{371F} & 11 & 2^8\cdot 157\cdot 76723322773093\\
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16 & \text{257B} & 14 & 2^7\cdot 29\cdot479\cdot71711\cdot
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409177\cdot654233\\
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\hline\end{array}$$
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\end{center}
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