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\begin{document}
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\noindent{\bf Tables to make it clear what we computed using modular methods.}
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Pages 1--2 contain a table which interleaves the modular and non-modular computations.
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Page 3 contains just the modular computations.\vspace{2ex}
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\noindent{\bf Modular computations for optimal quotients. }
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The rows labeled with $N$ were computed by Flynn, Lepr\'{e}vost, Schaefer,
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Stoll, and Wetherell and refer to the curves of Table 1 of {\em Empirical evidence}.
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The rows labeled in bold below $A$ were computed by me (Stein)
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using ``modular algorithms,'' and refer to the corresponding
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optimal quotients of $J_0(N)$. There are exactly four cases for
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which our computations definitely do not agree: these are the four
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Hasegawa (but not Wang) curves. This is fine because these Hasegawa
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curves are not optimal.
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The notation $a|T|b$ means that $a$ divides $T$ and $T$ divides $b$.
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\vspace{3ex}
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\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}
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\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}
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\begin{center}
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\begin{tabular}{|l|l|c|r@{.}l|r@{.}l|l|l|c|c|}
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\hline
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\mcc{$A$} & \mcc{$N$} & $r$
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& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
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& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & T & \mcc{disagreement}
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\\ \hline\hline
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% A N r L(1) Om Reg cp's Tor
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& 23 & 0 & 0&24843 & 2&7328 & 1 & 11 & 11 & \\
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{\bf 23A} & & & 0&24843 & 2&7328 & 1 & 11 & 11 & \\
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& 29 & 0 & 0&29152 & 2&0407 & 1 & 7 & 7 & \\
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{\bf 29A} & & - & 0&29152 & 2&0407 & - & 7 & 7 & \\
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& 31 & 0 & 0&44929 & 2&2464 & 1 & 5 & 5 & \\
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{\bf 31A} & & - & 0&44929 & 2&2464 & - & 5 & 5 & \\
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\hline
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& 35 & 0 & 0&37275 & 2&9820 & 1 & 16,2 & 16 & \\
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{\bf 35B} & & - & 0&37275 & 2&9820 & - & 8,$2|c_7 | 4$ & $8|T|16$ & *\\
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& 39 & 0 & 0&38204 & 10&697 & 1 & 28,1 & 28 & \\
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{\bf 39B} & & - & 0&38204 & 10&697 & - & 14,2 & $14|T|28$ & *\\
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& 63 & 0 & 0&75328 & 4&5197 & 1 & 2,3 & 6 & \\
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{\bf 63B} & & - & 0&75328 & 4&5197 & - & ?,3 & $6|T|12$ &\\
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& 65,A & 0 & 0&45207 & 6&3289 & 1 & 7,1 & 14 & \\
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{\bf 65C} & & - & 0&45207 & 6&3289 & - & 7,1 & $7|T|14$& \\
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& 65,B & 0 & 0&91225 & 5&4735 & 1 & 1,3 & 6 & \\
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{\bf 65B} & & - & 0&91225 & 5&4735 & - & 1,3 & $3|T|6$ & \\
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\hline
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& 67 & 2 & 0&23410 & 20&465 & 0.011439 & 1 & 1 & \\
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{\bf 67B} & & & & & 20&465 & - & 1 & 1 & \\
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& 73 & 2 & 0&25812 & 24&093 & 0.010713 & 1 & 1 & \\
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{\bf 73B} & & & & & 24&093 & - & 1 & 1 & \\
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& 85 & 2 & 0&34334 & 9&1728 & 0.018715 & 4,2 & 2 & \\
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{\bf 85B} & & & & & 18&3455 & - & 2,1 & $1|T|2$ & *\\
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& 87 & 0 & 1&4323 & 7&1617 & 1 & 5,1 & 5 & \\
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{\bf 87A} & & & 1&4323 & 7&1617 & - & 5,1 & 5 & \\
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\hline
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& 93 & 2 & 0&33996 & 18&142 & 0.0046847 & 4,1 & 1 & \\
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{\bf 93A} & & & & & 18&142 & - & $2|c_3|4$,1 & 1 & \\
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& 103 & 2 & 0&37585 & 16&855 & 0.022299 & 1 & 1 & \\
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{\bf 103A} & & & & & 16&855 & - & 1 & 1 & \\
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& 107 & 2 & 0&53438 & 11&883 & 0.044970 & 1 & 1 & \\
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{\bf 107A} & & & & & 11&883 & - & 1 & 1 & \\
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& 115 & 2 & 0&41693 & 10&678 & 0.0097618 & 4,1 & 1 & \\
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{\bf 115B} & & & & & 10&678 & - & $2|c_5|4$, 1 & 1 & \\
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\hline
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\end{tabular}
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\vfill
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\mbox{}
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\begin{tabular}{|l|l|c|r@{.}l|r@{.}l|l|l|c|c|}
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\hline
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\mcc{$A$} & \mcc{$C$} & $r$
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& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
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& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $|\text{T}|$ & \mcc{disagreement}
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\\ \hline\hline
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% A N r L(1) Om Reg cp's Tor
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& 117,A & 0 & 1&0985 & 3&2954 & 1 & 4,3 & 6 & \\
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{\bf 117B} & & & 1&0985 & 3&2954 & & ?,3 & $3|T|12$ & \\
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& 117,B & 0 & 1&9510 & 1&9510 & 1 & 4,1 & 2 & \\
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{\bf 117C} & & & 1&9510 & 1&9510 & & ?,1 & $1|T|2^2$ & \\
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& 125,A & 2 & 0&62996 & 13&026 & 0.048361 & 1 & 1 & \\
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{\bf 125A} & & & & & 13&026 & - & ? & 1 & \\
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& 125,B & 0 & 2&0842 & 2&6052 & 1 & 5 & 5 & \\
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{\bf 125B} & & & 2&0842 & 2&6052 & - & ? & 5 & \\
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\hline
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& 133,A & 0 & 2&2265 & 2&7832 & 1 & 5,1 & 5 & \\
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{\bf 133B} & & & 2&2265 & 2&7832 & - & 5,1 & 5 & \\
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& 133,B & 2 & 0&43884 & 15&318 & 0.028648 & 1,1 & 1 & \\
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{\bf 133A} & & & & & 15&318 & & 1,1 & 1 & \\
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& 135 & 0 & 1&5110 & 4&5331 & 1 & 3,1 & 3 & \\
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{\bf 135D} & & & 1&5110 & 4&5331 & - & ?,1 & 3 & \\
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& 147 & 2 & 0&61816 & 13&616 & 0.045400 & 2,2 & 2 & \\
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{\bf 147D} & & & & & 13&616 & - & ?,2 & $1|T|2$ & \\
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\hline
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& 161 & 2 & 0&82364 & 11&871 & 0.017345 & 4,1 & 1 & \\
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{\bf 161B} & & & & & 11&871 & - & $2|T|4$,1 & 1 & \\
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& 165 & 2 & 0&68650 & 9&5431 & 0.071936 & 4,2,2 & 4 & \\
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{\bf 165A} & & & & & 9&5431 & - & 2,1,1 & $1|T|4$ & *\\
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& 167 & 2 & 0&91530 & 7&3327 & 0.12482 & 1 & 1 & \\
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{\bf 167A} & & & & & 7&3327 & - & 1 & 1 & \\
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& 175 & 0 & 0&97209 & 4&8605 & 1 & 1,5 & 5 & \\
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{\bf 175E} & & & 0&972 & 4&86 & & ?,5 & 5 & \\
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\hline
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& 177 & 2 & 0&90451 & 13&742 & 0.065821 & 1,1 & 1 & \\
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{\bf 177A} & & & & & 13&742 & - & 1,1 & 1 & \\
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& 188 & 2 & 1&1708 & 11&519 & 0.011293 & 9,1 & 1 & \\
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{\bf 188B} & & & & & 11&519 & - & ?,1 & 1 & \\
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& 189 & 0 & 1&2982 & 3&8946 & 1 & 1,3 & 3 & \\
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{\bf 189E} & & & 1&2982 & 3&894 & - &?,3 & 3 & \\
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& 191 & 2 & 0&95958 & 17&357 & 0.055286 & 1 & 1 & \\
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{\bf 191A} & & & & & 17&35 & & 1 & 1 & \\
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\hline
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\end{tabular}
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\end{center}
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\newpage
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\noindent
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{\bf Part of table computed using modular algorithms.}
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For clarity, here is just the part of the table which can be computed using only
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modular algorithms. Vanishing or not of $L(J,1)$ was also
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computed and agreed in all cases. I have no information about
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the $c_p$ when $p^2\mid N$. My understanding is that the value
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$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$
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was also computed by Joe (?) using modular symbols. I haven't checked
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this computation when $r>0$ because I haven't implimented the algorithm.
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In all but 6 of the optimal cases, the exact torsion subgroup of the optimal
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quotient could be determined using Hecke operators and modular symbols
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(we don't mention this in the paper).
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\vspace{2ex}
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\begin{center}
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\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|}
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\hline
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\mcc{$A$} & $r$
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& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
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& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & T
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\\ \hline\hline
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% A r L(1) Om Reg cp's Tor
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{\bf 23A} & & 0&24843 & 2&7328 & 1 & 11 & 11 \\
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{\bf 29A} & - & 0&29152 & 2&0407 & - & 7 & 7 \\
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{\bf 31A} & - & 0&44929 & 2&2464 & - & 5 & 5 \\
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\hline
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{\bf 35B} & - & 0&37275 & 2&9820 & - & 8,$2|c_7 | 4$ & $8|T|16$ \\
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{\bf 39B} & - & 0&38204 & 10&697 & - & 14,2 & $14|T|28$ \\
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{\bf 63B} & - & 0&75328 & 4&5197 & - & ?,3 & $6|T|12$\\
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{\bf 65C} & - & 0&45207 & 6&3289 & - & 7,1 & $7|T|14$\\
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{\bf 65B} & - & 0&91225 & 5&4735 & - & 1,3 & $3|T|6$ \\
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\hline
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{\bf 67B} & & & & 20&465 & - & 1 & 1 \\
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{\bf 73B} & & & & 24&093 & - & 1 & 1 \\
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{\bf 85B} & & & & 18&3455 & - & 2,1 & $1|T|2$ \\
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{\bf 87A} & & 1&4323 & 7&1617 & - & 5,1 & 5 \\
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\hline
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{\bf 93A} & & & & 18&142 & - & $2|c_3|4$,1 & 1 \\
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{\bf 103A} & & & & 16&855 & - & 1 & 1 \\
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{\bf 107A} & & & & 11&883 & - & 1 & 1 \\
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{\bf 115B} & & & & 10&678 & - & $2|c_5|4$, 1 & 1 \\
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\hline
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{\bf 117B} & & 1&0985 & 3&2954 & & ?,3 & $3|T|12$ \\
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{\bf 117C} & & 1&9510 & 1&9510 & & ?,1 & $1|T|2^2$ \\
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{\bf 125A} & & & & 13&026 & - & ? & 1 \\
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{\bf 125B} & & 2&0842 & 2&6052 & - & ? & 5 \\
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\hline
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{\bf 133B} & & 2&2265 & 2&7832 & - & 5,1 & 5 \\
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{\bf 133A} & & & & 15&318 & & 1,1 & 1 \\
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{\bf 135D} & & 1&5110 & 4&5331 & - & ?,1 & 3 \\
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{\bf 147D} & & & & 13&616 & - & ?,2 & $1|T|2$ \\
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\hline
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{\bf 161B} & & & & 11&871 & - & $2|T|4$,1 & 1 \\
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{\bf 165A} & & & & 9&5431 & - & 2,1,1 & $1|T|4$ \\
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{\bf 167A} & & & & 7&3327 & - & 1 & 1 \\
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{\bf 175E} & & 0&972 & 4&86 & & ?,5 & 5 \\
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\hline
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{\bf 177A} & & & & 13&742 & - & 1,1 & 1\\
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{\bf 188B} & & & & 11&519 & - & ?,1 & 1 \\
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{\bf 189E} & & 1&2982 & 3&894 & - &?,3 & 3 \\
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{\bf 191A} & & & & 17&35 & & 1 & 1 \\
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\hline
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\end{tabular}
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\end{center}
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\end{document}
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