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\begin{document}
\noindent{\bf Tables to make it clear what we computed using modular methods.}
Pages 1--2 contain a table which interleaves the modular and non-modular computations.
Page 3 contains just the modular computations.\vspace{2ex}

\noindent{\bf Modular computations for optimal quotients. }
The rows labeled with $N$ were computed by Flynn, Lepr\'{e}vost, Schaefer,
Stoll, and Wetherell and refer to the curves of Table 1 of {\em Empirical evidence}.
The rows labeled in bold below $A$ were computed by me (Stein) 
using ``modular algorithms,'' and refer to the corresponding 
optimal quotients of $J_0(N)$.   There are exactly four cases for 
which our computations definitely do not agree: these are the four 
Hasegawa  (but not Wang) curves.  This is fine because these Hasegawa
curves are not optimal. 
The notation $a|T|b$ means that $a$ divides $T$ and $T$ divides $b$.
\vspace{3ex}

\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}
\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}
\begin{center}
\begin{tabular}{|l|l|c|r@{.}l|r@{.}l|l|l|c|c|}
\hline
\mcc{$A$} & \mcc{$N$} & $r$
& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & T & \mcc{disagreement} 
\\ \hline\hline
%  A        N        r   L(1)       Om       Reg         cp's   Tor  
           & 23    & 0 & 0&24843 &  2&7328 & 1         & 11    & 11 & \\
{\bf 23A}  &       &   & 0&24843 &  2&7328 & 1         & 11    & 11 & \\
           & 29    & 0 & 0&29152 &  2&0407 & 1         & 7     &  7 & \\
{\bf 29A}  &       & - & 0&29152 &  2&0407 & -         & 7     &  7 & \\
           & 31    & 0 & 0&44929 &  2&2464 & 1         & 5     &  5 &  \\
{\bf 31A}  &       & - & 0&44929 &  2&2464 & -         & 5     &  5 & \\
\hline
           & 35    & 0 & 0&37275 &  2&9820 & 1         & 16,2  & 16  & \\ 
{\bf 35B}  &       & - & 0&37275 &  2&9820 & -         & 8,$2|c_7 | 4$   & $8|T|16$   & *\\
           & 39    & 0 & 0&38204 & 10&697  & 1         & 28,1  & 28 &  \\
{\bf 39B}  &       & - & 0&38204 & 10&697 & -         & 14,2  & $14|T|28$  &  *\\
           & 63    & 0 & 0&75328 &  4&5197 & 1         & 2,3   &  6  &  \\
{\bf 63B}  &       & - & 0&75328 &  4&5197 & -         & ?,3   &  $6|T|12$ &\\
           & 65,A  & 0 & 0&45207 &  6&3289 & 1         & 7,1   & 14 &  \\
{\bf 65C}  &       & - & 0&45207 &  6&3289 & -         & 7,1   & $7|T|14$&   \\
           & 65,B  & 0 & 0&91225 &  5&4735 & 1         & 1,3   &  6  & \\ 
{\bf 65B}  &       & - & 0&91225 &  5&4735 & -         & 1,3   & $3|T|6$ &   \\
\hline
           & 67    & 2 & 0&23410 & 20&465  & 0.011439  & 1     &  1  & \\
{\bf 67B}  &       &   &  &      & 20&465  & -         & 1     &  1  & \\
           & 73    & 2 & 0&25812 & 24&093  & 0.010713  & 1     &  1  &  \\
{\bf 73B}  &       &   &  &      & 24&093  & -         & 1     &  1 & \\
           & 85    & 2 & 0&34334 &  9&1728 & 0.018715  & 4,2   &  2  &  \\
{\bf 85B}  &       &   &  &      & 18&3455 & -         & 2,1   &  $1|T|2$ & *\\
           & 87    & 0 & 1&4323  &  7&1617 & 1         & 5,1   &  5  &  \\ 
{\bf 87A}  &       &   & 1&4323  &  7&1617 & -         & 5,1   &  5  & \\
\hline
           & 93    & 2 & 0&33996 & 18&142  & 0.0046847 & 4,1   &  1  & \\
{\bf 93A}  &       &   &  &      & 18&142  & -         & $2|c_3|4$,1 &  1 & \\
           & 103   & 2 & 0&37585 & 16&855  & 0.022299  & 1     &  1  &  \\
{\bf 103A} &       &   &  &      & 16&855  & -         & 1     &  1  & \\
           & 107   & 2 & 0&53438 & 11&883  & 0.044970  & 1     &  1  &  \\
{\bf 107A} &       &   &  &      & 11&883  & -         & 1     &  1  & \\
           & 115   & 2 & 0&41693 & 10&678  & 0.0097618 & 4,1             &  1  &  \\
{\bf 115B} &       &   &  &      & 10&678  & -         & $2|c_5|4$, 1    & 1  & \\
\hline
\end{tabular}
\vfill
\mbox{}

\begin{tabular}{|l|l|c|r@{.}l|r@{.}l|l|l|c|c|}
\hline
\mcc{$A$} & \mcc{$C$} & $r$
& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $|\text{T}|$ & \mcc{disagreement} 
\\ \hline\hline
%  A        N        r   L(1)       Om       Reg         cp's   Tor  
            & 117,A & 0 & 1&0985  &  3&2954 & 1         & 4,3   &  6  &  \\
{\bf 117B}  &       &   & 1&0985  &  3&2954 &           & ?,3   & $3|T|12$  & \\
            & 117,B & 0 & 1&9510  &  1&9510 & 1         & 4,1   &  2  & \\
{\bf 117C}  &       &   & 1&9510  &  1&9510 &           & ?,1   & $1|T|2^2$ & \\
            & 125,A & 2 & 0&62996 & 13&026  & 0.048361  & 1     &  1  &  \\
{\bf 125A}  &       &   &  &      & 13&026  & -         & ?     &  1 & \\
            & 125,B & 0 & 2&0842  &  2&6052 & 1         & 5     &  5  &  \\
{\bf 125B}  &       &   & 2&0842  &  2&6052 & -         & ?     &  5 & \\
\hline
            & 133,A & 0 & 2&2265  &  2&7832 & 1         & 5,1   &  5  &  \\
{\bf 133B}  &       &   & 2&2265  &  2&7832 & -         & 5,1   &  5  & \\
            & 133,B & 2 & 0&43884 & 15&318  & 0.028648  & 1,1   &  1  &  \\
{\bf 133A}  &       &   &  &      & 15&318  &           & 1,1   &  1 & \\
            & 135   & 0 & 1&5110  &  4&5331 & 1         & 3,1   &  3  &  \\
{\bf 135D}  &       &   & 1&5110  &  4&5331 & -         & ?,1   &  3 & \\
            & 147   & 2 & 0&61816 & 13&616  & 0.045400  & 2,2   &  2  &  \\
{\bf 147D}  &       &   &  &      & 13&616  & -         & ?,2   &  $1|T|2$ & \\
\hline
            & 161   & 2 & 0&82364 & 11&871  & 0.017345  & 4,1   &  1  &  \\
{\bf 161B}  &       &   &  &      & 11&871  & -         & $2|T|4$,1  & 1   & \\
            & 165   & 2 & 0&68650 &  9&5431 & 0.071936  & 4,2,2 &  4  &  \\
{\bf 165A}  &       &   &  &      &  9&5431 & -         & 2,1,1 &  $1|T|4$ & *\\
            & 167   & 2 & 0&91530 &  7&3327 & 0.12482   & 1     &  1  &  \\
{\bf 167A}  &       &   &  &      &  7&3327 & -         & 1     &  1  & \\
            & 175   & 0 & 0&97209 &  4&8605 & 1         & 1,5   &  5  &  \\
{\bf 175E}  &       &   & 0&972   &  4&86     &           & ?,5   &  5  & \\
\hline
            & 177   & 2 & 0&90451 & 13&742  & 0.065821  & 1,1   &  1 &  \\
{\bf 177A}  &       &   &  &      & 13&742  & -         & 1,1    & 1 & \\
            & 188   & 2 & 1&1708  & 11&519  & 0.011293  & 9,1   &  1 &  \\
 {\bf 188B} &       &   &  &      & 11&519  & -         & ?,1      & 1  & \\
            & 189   & 0 & 1&2982  &  3&8946 & 1         & 1,3   &  3 &  \\
 {\bf 189E} &       &   & 1&2982  &  3&894  & -          &?,3    & 3  & \\
            & 191   & 2 & 0&95958 & 17&357  & 0.055286  & 1     &  1 &  \\
 {\bf 191A} &       &   &  &      & 17&35   &           & 1     &  1 & \\
\hline
\end{tabular}
\end{center}

\newpage
\noindent 
{\bf Part of table computed using modular algorithms.}
For clarity, here is just the part of the table which can be computed using only 
modular algorithms.  Vanishing or not of $L(J,1)$ was also
computed and agreed in all cases.  I have no information about
the $c_p$ when $p^2\mid N$.  My understanding is that the value 
$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$
was also computed by Joe (?) using modular symbols.  I haven't checked
this computation when $r>0$ because I haven't implimented the algorithm.
In all but 6 of the optimal cases, the exact torsion subgroup of the optimal
quotient could be determined using Hecke operators and modular symbols
(we don't mention this in the paper). 
\vspace{2ex}

\begin{center}
\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|}
\hline
\mcc{$A$} & $r$
& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & T 
\\ \hline\hline
%  A               r   L(1)       Om       Reg         cp's   Tor  
{\bf 23A}         &   & 0&24843 &  2&7328 & 1         & 11    & 11 \\
{\bf 29A}         & - & 0&29152 &  2&0407 & -         & 7     &  7 \\
{\bf 31A}         & - & 0&44929 &  2&2464 & -         & 5     &  5 \\
\hline
{\bf 35B}         & - & 0&37275 &  2&9820 & -         & 8,$2|c_7 | 4$   & $8|T|16$  \\
{\bf 39B}         & - & 0&38204 & 10&697 & -         & 14,2  & $14|T|28$ \\
{\bf 63B}         & - & 0&75328 &  4&5197 & -         & ?,3   &  $6|T|12$\\
{\bf 65C}         & - & 0&45207 &  6&3289 & -         & 7,1   & $7|T|14$\\
{\bf 65B}         & - & 0&91225 &  5&4735 & -         & 1,3   & $3|T|6$ \\
\hline
{\bf 67B}         &   &  &      & 20&465  & -         & 1     &  1  \\
{\bf 73B}         &   &  &      & 24&093  & -         & 1     &  1 \\
{\bf 85B}         &   &  &      & 18&3455 & -         & 2,1   &  $1|T|2$ \\
{\bf 87A}         &   & 1&4323  &  7&1617 & -         & 5,1   &  5  \\
\hline
{\bf 93A}        &   &  &      & 18&142  & -         & $2|c_3|4$,1 &  1 \\
{\bf 103A}        &   &  &      & 16&855  & -         & 1     &  1  \\
{\bf 107A}        &   &  &      & 11&883  & -         & 1     &  1  \\
{\bf 115B}        &   &  &      & 10&678  & -         & $2|c_5|4$, 1    & 1  \\
\hline
{\bf 117B}         &   & 1&0985  &  3&2954 &           & ?,3   & $3|T|12$  \\
{\bf 117C}         &   & 1&9510  &  1&9510 &           & ?,1   & $1|T|2^2$ \\
{\bf 125A}         &   &  &      & 13&026  & -         & ?     &  1 \\
{\bf 125B}         &   & 2&0842  &  2&6052 & -         & ?     &  5 \\
\hline
{\bf 133B}         &   & 2&2265  &  2&7832 & -         & 5,1   &  5  \\
{\bf 133A}         &   &  &      & 15&318  &           & 1,1   &  1 \\
{\bf 135D}         &   & 1&5110  &  4&5331 & -         & ?,1   &  3  \\
{\bf 147D}         &   &  &      & 13&616  & -         & ?,2   &  $1|T|2$ \\
\hline
{\bf 161B}         &   &  &      & 11&871  & -         & $2|T|4$,1  & 1   \\
{\bf 165A}         &   &  &      &  9&5431 & -         & 2,1,1 &  $1|T|4$ \\
{\bf 167A}         &   &  &      &  7&3327 & -         & 1     &  1  \\
{\bf 175E}         &   & 0&972   &  4&86     &           & ?,5   &  5   \\
\hline
{\bf 177A}         &   &  &      & 13&742  & -         & 1,1    & 1\\
 {\bf 188B}        &   &  &      & 11&519  & -         & ?,1      & 1  \\
 {\bf 189E}        &   & 1&2982  &  3&894  & -          &?,3    & 3  \\
 {\bf 191A}        &   &  &      & 17&35   &           & 1     &  1  \\
\hline
\end{tabular}
\end{center}


\end{document}