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.

Modular computations for optimal quotients. The rows labeled with N were computed by Flynn, Leprévost, Schaefer, Stoll, and Wetherell and refer to the curves of Table 1 of 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.

A  N r lim_{s→1} ^{\frac{L(J,s)}{(s−1)^{r}}}  Ω  Reg  c_{p}’s  T  disagreement
23 0 0.24843  2.7328 1  11  11
23A  0.24843  2.7328 1  11  11
29 0 0.29152  2.0407 1  7  7
29A  - 0.29152  2.0407 -  7  7
31 0 0.44929  2.2464 1  5  5
31A  - 0.44929  2.2464 -  5  5
35 0 0.37275  2.9820 1  16,2  16
35B  - 0.37275  2.9820 -  8,2|c_{7}|4 8|T |16  *
39 0 0.38204 10.697 1  28,1  28
39B  - 0.38204 10.697 -  14,2  14|T |28  *
63 0 0.75328  4.5197 1  2,3  6
63B  - 0.75328  4.5197 -  ?,3  6|T |12
65,A 0 0.45207  6.3289 1  7,1  14
65C  - 0.45207  6.3289 -  7,1  7|T |14
65,B 0 0.91225  5.4735 1  1,3  6
65B  - 0.91225  5.4735 -  1,3  3|T |6
67 2 0.23410 20.465 0.011439 1  1
67B . 20.465 - 1 1
73 2 0.25812 24.093 0.010713 1  1
73B  .  24.093 -  1  1
85 2 0.34334  9.1728 0.018715 4,2  2
85B  .  18.3455 -  2,1  1|T |2  *
87 0 1.4323  7.1617 1  5,1  5
87A  1.4323  7.1617 -  5,1  5
93 2 0.33996 18.142 0.0046847 4,1  1
93A . 18.142 - 2|c_{3}|4,1 1
103 2 0.37585 16.855 0.022299 1  1
103A . 16.855 - 1 1
107 2 0.53438 11.883 0.044970 1  1
107A . 11.883 - 1 1
115 2 0.41693 10.678 0.0097618 4,1  1
115B  .  10.678 -  2|c_{5}|4, 1  1

A  C r lim_{s→1} ^{\frac{L(J,s)}{(s−1)^{r}}}  Ω  Reg  c_{p}’s  |T| disagreement
117,A 0 1.0985  3.2954 1  4,3  6
117B  1.0985  3.2954  ?,3  3|T |12
117,B 0 1.9510  1.9510 1  4,1  2
117C  1.9510  1.9510  ?,1  1|T |2^{2}
125,A 2 0.62996 13.026 0.048361 1  1
125A  .  13.026 -  ?  1
125,B 0 2.0842  2.6052 1  5  5
125B  2.0842  2.6052 -  ?  5
133,A 0 2.2265  2.7832 1  5,1  5
133B  2.2265  2.7832 -  5,1  5
133,B 2 0.43884 15.318 0.028648 1,1  1
133A  .  15.318  1,1  1
135 0 1.5110  4.5331 1  3,1  3
135D  1.5110  4.5331 -  ?,1  3
147 2 0.61816 13.616 0.045400 2,2  2
147D . 13.616 - ?,2 1|T |2
161 2 0.82364 11.871 0.017345 4,1  1
161B  .  11.871 -  2|T |4,1 1
165 2 0.68650  9.5431 0.071936 4,2,2  4
165A  .  9.5431 -  2,1,1  1|T |4  *
167 2 0.91530  7.3327 0.12482 1  1
167A  .  7.3327 -  1  1
175 0 0.97209  4.8605 1  1,5  5
175E  0.972  4.86  ?,5  5
177 2 0.90451 13.742 0.065821 1,1  1
177A  .  13.742 -  1,1  1
188 2 1.1708  11.519 0.011293 9,1  1
188B  .  11.519 -  ?,1  1
189 0 1.2982  3.8946 1  1,3  3
189E  1.2982  3.894 -  ?,3  3
191 2 0.95958 17.357 0.055286 1  1
191A  .  17.35  1  1

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} | N. My understanding is that the value lim_{s→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).

A r lim_{s→1} ^{\frac{L(J,s)}{(s−1)^{r}}}  Ω  Reg c_{p}’s  T
23A  0.24843  2.7328 1 11  11
29A - 0.29152  2.0407 -  7  7
31A - 0.44929  2.2464 -  5  5
35B - 0.37275  2.9820 -  8,2|c_{7}|4 8|T |16
39B - 0.38204 10.697 -  14,2  14|T |28
63B - 0.75328  4.5197 -  ?,3  6|T |12
65C - 0.45207  6.3289 -  7,1  7|T |14
65B - 0.91225  5.4735 -  1,3  3|T |6
67B  .  20.465 -  1  1
73B  .  24.093 -  1  1
85B  .  18.3455 -  2,1  1|T |2
87A  1.4323  7.1617 -  5,1  5
93A  .  18.142 -  2|c_{3}|4,1  1
103A  .  16.855 -  1  1
107A  .  11.883 -  1  1
115B  .  10.678 -  2|c_{5}|4, 1  1
117B  1.0985  3.2954  ?,3  3|T |12
117C  1.9510  1.9510  ?,1  1|T |2^{2}
125A  .  13.026 -  ?  1
125B  2.0842  2.6052 -  ?  5
133B  2.2265  2.7832 -  5,1  5
133A  .  15.318  1,1  1
135D  1.5110  4.5331 -  ?,1  3
147D  .  13.616 -  ?,2  1|T |2
161B  .  11.871 -  2|T |4,1  1
165A  .  9.5431 -  2,1,1  1|T |4
167A  .  7.3327 -  1  1
175E  0.972  4.86  ?,5  5
177A  .  13.742 -  1,1  1
188B  .  11.519 -  ?,1  1
189E  1.2982  3.894 -  ?,3  3
191A  .  17.35  1  1