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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{Some genus two quotients of $X_1(N)$}
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\author{William A. Stein}
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\date{June 21, 2000}
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\bibliographystyle{amsplain}
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\begin{document}
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\maketitle
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\begin{abstract}
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The author has written a program to compute in $S_2(\Gamma_1(N))$.
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As an application, we use the method described clearly in
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Chapter~4 of \cite{galbraith} to compute models for
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genus two quotients of $X_1(N)$ by Atkin-Lehner involutions.
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I don't know if somebody else has already made a table like this,
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but if they did they may have made mistakes so it is good to recompute it.
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\end{abstract}
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\section*{Introduction}
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The following is a model for $X_1(13)$:
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$$ y^2 = x^6 + 2x^5 + x^4 + 2x^3 + 6x^2 + 4x + 1 .$$
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\comment{
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It was found using the \hecke{} package in \magma{}, as follows.
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\begin{verbatim}
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> G<eps>:=DirichletGroup(13,CyclotomicField(12));
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> A:=CS(MS(eps^2,2));
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> B:=CS(MS(eps^10,2)); // The Galois conjugate...
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> qEigenform(A,13);
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q + (-zeta_12^2 - 1)*q^2 + (2*zeta_12^2 - 2)*q^3 + zeta_12^2*q^4
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+ (-2*zeta_12^2 + 1)*q^5 + (-2*zeta_12^2 + 4)*q^6 +
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(2*zeta_12^2 - 1)*q^8 - zeta_12^2*q^9 + (3*zeta_12^2 -
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3)*q^10 - 2*q^12 + O(q^13)
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> qEigenform(B,13);
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q + (zeta_12^2 - 2)*q^2 - 2*zeta_12^2*q^3 + (-zeta_12^2 + 1)*q^4
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+ (2*zeta_12^2 - 1)*q^5 + (2*zeta_12^2 + 2)*q^6 +
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(-2*zeta_12^2 + 1)*q^8 + (zeta_12^2 - 1)*q^9 -
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3*zeta_12^2*q^10 - 2*q^12 + O(q^13)
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> E:=[Eltseq(Coefficient(qEigenform(A,38),n)): n in [1..37]];
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> R<q>:=LaurentSeriesRing(Rationals());
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> f:=&+[E[n][1]*q^n : n in [1..37]]];
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> g:=-&+[E[n][3]*q^n : n in [1..37]];
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> f;
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q - q^2 - 2*q^3 + q^5 + 4*q^6 - q^8 - 3*q^10 - 2*q^12 - q^13 +
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2*q^15 + 5*q^16 - q^18 - 4*q^19 + 2*q^20 + 6*q^23 - 2*q^24 +
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2*q^25 - 2*q^26 - 4*q^27 - 3*q^29 - 2*q^31 - 6*q^32 + 3*q^34
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+ q^36 + 5*q^37
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> g;
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q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13
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- 2*q^15 + 5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + q^20 + 6*q^23
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+ 2*q^24 - 7*q^26 - 3*q^29 + 6*q^30 - 4*q^31 - 3*q^32 +
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6*q^34 + q^36 - 5*q^37
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> X:=f/g;
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> X;
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q^-1 + 1 + q + q^2 + q^4 - q^6 - 2*q^8 - q^11 + 2*q^12 + 2*q^14 +
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3*q^15 - q^16 + 2*q^17 - 2*q^18 - 3*q^19 - 6*q^21 - q^22 -
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q^23 - 2*q^24 + 7*q^25 + q^26 + 5*q^27 + 7*q^28 - 2*q^29 +
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6*q^30 - 4*q^31 - 7*q^32 - q^33 - 14*q^34 - q^35 + O(q^36)
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> Y:=q*Derivative(X)/g;
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> Y;
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-q^-3 - 2*q^-2 - 4*q^-1 - 6 - 10*q - 10*q^2 - 12*q^3 - 11*q^4
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- 3*q^5 - 3*q^6 + 11*q^7 + 22*q^8 + 18*q^9 + 36*q^10
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+ 12*q^11 + q^12 - 3*q^13 - 65*q^14 - 44*q^15 - 78*q^16
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- 90*q^17 + 10*q^18 - 33*q^19 + 100*q^20 + 165*q^21
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+ 101*q^22 + 290*q^23 + 76*q^24 + 33*q^25 + 28*q^26
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- 435*q^27 - 208*q^28 - 475*q^29 - 558*q^30 + 62*q^31
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- 331*q^32 + 511*q^33 + O(q^34)
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> Y^2-X^6;
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-2*q^-5 - 9*q^-4 + ...
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> Y^2-X^6+2*X^5;
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q^-4 + 2*q^-3 + ...
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// etc.
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> Y^2-X^6+2*X^5-X^4+2*X^3-6*X^2+4*X-1; // model for X_1(13)!!!
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O(q^31)
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\end{verbatim}}
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\section{Spaces of cusp forms}
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Observe that
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$$S_2(\Gamma_1(N)) = \bigoplus_{\chi:(\Z/N\Z)^*\ra\C^*} S_2(N,\chi),$$
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where $S_2(N,\chi)$ is the space of cusp forms on which the
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diamond operators $\langle d\rangle$ act via the character~$\chi$.
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For efficiency purposes, we compute only the spaces $S_2(N,\chi)$.
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Notationaly, we represent
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a character~$\chi$ is represented as follows, with appropriate modifications
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at~$2$. Write $(\Z/N\Z)^*$ as a product of cyclic groups corresponding
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to the prime powers dividing~$N$. Let $\eps_1,\ldots, \eps_n$ be characters
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that send minimal generators of these cyclic factors to appropriate roots
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of unity in~$\C^*$. We write characters $\chi$ in terms of the $\eps_i$.
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The dimension of $S_2(\Gamma_1(N))$ equals the genus of
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$X_1(N)$, which is a number that is recorded in
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Table~\ref{table:genus} for $N\leq 100$.
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\begin{table}
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\begin{center}
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\caption{The genus of $X_1(N)$\label{table:genus}}
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\vspace{1ex}
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\begin{tabular}{|lc|}\hline
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$N$ & $g$\\
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$1$ & $0$ \\
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$2$ & $0$ \\
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$3$ & $0$ \\
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$4$ & $0$ \\
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$5$ & $0$ \\
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$6$ & $0$ \\
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$7$ & $0$ \\
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$8$ & $0$ \\
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$9$ & $0$ \\
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$10$ & $0$ \\
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$11$ & $1$ \\
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$12$ & $0$ \\
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$13$ & $2$ \\
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$14$ & $1$ \\
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$15$ & $1$ \\
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$16$ & $2$ \\
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$17$ & $5$ \\
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$18$ & $2$ \\
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$19$ & $7$ \\
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$20$ & $3$ \\\hline
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\end{tabular}
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\quad
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\begin{tabular}{|lc|}\hline
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$N$ & $g$\\
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$21$ & $5$ \\
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$22$ & $6$ \\
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$23$ & $12$ \\
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$24$ & $5$ \\
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$25$ & $12$ \\
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$26$ & $10$ \\
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$27$ & $13$ \\
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$28$ & $10$ \\
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$29$ & $22$ \\
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$30$ & $9$ \\
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$31$ & $26$ \\
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$32$ & $17$ \\
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$33$ & $21$ \\
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$34$ & $21$ \\
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$35$ & $25$ \\
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$36$ & $17$ \\
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$37$ & $40$ \\
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$38$ & $28$ \\
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$39$ & $33$ \\
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$40$ & $25$ \\\hline
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\end{tabular}
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\quad
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\begin{tabular}{|lc|}\hline
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$N$ & $g$\\
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$41$ & $51$ \\
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$42$ & $25$ \\
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$43$ & $57$ \\
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$44$ & $36$ \\
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$45$ & $41$ \\
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$46$ & $45$ \\
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$47$ & $70$ \\
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$48$ & $37$ \\
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$49$ & $69$ \\
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$50$ & $48$ \\
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$51$ & $65$ \\
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$52$ & $55$ \\
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$53$ & $92$ \\
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$54$ & $52$ \\
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$55$ & $81$ \\
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$56$ & $61$ \\
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$57$ & $85$ \\
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$58$ & $78$ \\
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$59$ & $117$ \\
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$60$ & $57$ \\\hline
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\end{tabular}
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\quad
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\begin{tabular}{|lc|}\hline
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$N$ & $g$\\
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$61$ & $126$ \\
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$62$ & $91$ \\
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$63$ & $97$ \\
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$64$ & $93$ \\
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$65$ & $121$ \\
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$66$ & $81$ \\
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$67$ & $155$ \\
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$68$ & $105$ \\
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$69$ & $133$ \\
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$70$ & $97$ \\
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$71$ & $176$ \\
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$72$ & $97$ \\
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$73$ & $187$ \\
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$74$ & $136$ \\
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$75$ & $145$ \\
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$76$ & $136$ \\
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$77$ & $181$ \\
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$78$ & $121$ \\
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$79$ & $222$ \\
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$80$ & $137$ \\\hline
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\end{tabular}
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\quad
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\begin{tabular}{|lc|}\hline
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$N$ & $g$\\
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$81$ & $190$ \\
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$82$ & $171$ \\
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$83$ & $247$ \\
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$84$ & $133$ \\
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$85$ & $225$ \\
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$86$ & $190$ \\
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$87$ & $225$ \\
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$88$ & $181$ \\
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$89$ & $287$ \\
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$90$ & $153$ \\
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$91$ & $265$ \\
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$92$ & $210$ \\
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$93$ & $261$ \\
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$94$ & $231$ \\
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$95$ & $289$ \\
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$96$ & $193$ \\
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$97$ & $345$ \\
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$98$ & $235$ \\
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$99$ & $281$ \\
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$100$ & $231$ \\\hline
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\end{tabular}
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\end{center}
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\end{table}
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\section{Table of Genus 2 Curves}
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I do not know if somebody else has already made a table like this,
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but if they did they might have made mistakes so it is good that I
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have recomputed it.
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\setlength{\doublerulesep}{\arrayrulewidth}
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\begin{table}
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\begin{center}
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\caption{Genus 2 quotients $y^2=f(x)$ of $X_1(N)$\label{table:curves}
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that correspond to a single newform $f(q)$, {\bf
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I searched up to level $125$ and
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this is all I found!!}}
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\vspace{1ex}
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\begin{tabular}{|c|l|c|}\hline
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$X$, $\chi$ & $F(x)$, $f(q)$ & $L(J,1)\neq 0$?\\\hline\hline\hline
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&&\\
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$X_1(13)$ & $x^6 + 2x^5 + x^4 + 2x^3 + 6x^2 + 4x + 1 $ & yes\\
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&&\\\hline
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&&\\
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$X_1(16)$ & $x^6 - 2x^5 - x^4 - x^2 + 2x + 1$ & yes\\
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& $=(x - 1)(x + 1)(x^2 - 2x - 1)(x^2 + 1)$ & \\
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&&\\\hline
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&&\\
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$X_1(18)$ & $x^6 + 4x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1$ & yes\\
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\comment{&&\\\hline
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&&\\
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$X_0(22)$ & $x^6 - 4x^4 + 20x^3 - 40x^2 + 48x - 32$ & yes\\
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& $=(x^3 - 2x^2 + 4x - 4)(x^3 + 2x^2 - 4x + 8)$&\\
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&&\\\hline
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&&\\
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$X_0(23)$ & $x^6 - 8x^5 + 2x^4 + 2x^3 - 11x^2 + 10x - 7$ & yes\\
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& $=(x^3 - 8x^2 + 3x - 7)(x^3 - x + 1)$ &\\
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&&\\\hline
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&&\\
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$X_0(26)$ & $x^6 - 8x^5 + 8x^4 - 18x^3 + 8x^2 - 8x + 1$ & yes\\
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&&\\\hline
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&&\\
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$X_0(28)$ & $x^6 + 10x^4 + 25x^2 + 28$ & \\
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& $=(x^2 - x + 2)(x^2 + 7)(x^2 + x + 2)$ &\\
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&&\\\hline
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&&\\
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$X_0(29)$ & $x^6 - 4x^5 - 12x^4 + 2x^3 + 8x^2 + 8x - 7$ & \\
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&&\\\hline
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&&\\
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$X_0(30)/W_2$ & $x^6 + 2x^4 + 20x^3 - 19x^2 + 60x$ & \\
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& $=x(x + 3)(x^2 - 2*x + 5)(x^2 - x + 4)$\\}
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&&\\\hline
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&&\\ % char of order 2
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$X_1(45)/W$, $\eps_2^2$ & $x^6 + 22x^3 + 125$ & yes\\
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& $=(x^2 - 2x + 5)(x^4 + 2x^3 - x^2 + 10x + 25)$, &\\
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& $q + \sqrt{-5}q^2 - 3q^4 - \sqrt{-5}q^5 + \cdots$ & \\
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&&\\\hline
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\end{tabular}
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\end{center}
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\end{table}
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\newpage
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\bibliography{biblio}
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\end{document}