Higher Weight Data

1. The following are the valuations $d=d_4(\Gamma_0(p))$ at $p$ of the discriminant of the Hecke algebras associated to $S_4(\Gamma_0(p))$ for $p<500$.

   $p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
   $d$	0	0	0	0	0	2	2	2	2	4	4	6	6	6	6	8	8
   $p$	61	67	71	73	79	83	89	97	101	103	107	109	113	127	131	137	139
   $d$	10	10	10	12	12	12	14	16	16	16	16	18	18	20	20	22	24
   $p$	149	151	157	163	167	173	179	181	191	193	197	199	211	223	227	229	233
   $d$	24	24	26	26	26	28	28	30	30	32	32	32	34	36	36	38	38
   $p$	239	241	251	257	263	269	271	277	281	283	293	307	311	313	317	331	337
   $d$	38	40	40	42	42	44	44	46	46	46	48	50	50	52	52	54	56
   $p$	347	349	353	359	367	373	379	383	389	397	401	409	419	421	431	433	439
   $d$	56	58	58	58	60	62	62	62	65	66	66	68	68	70	70	72	72
   $p$	443	449	457	461	463	467	479	487	491	499
   $d$	72	74	76	76	76	76	78	80	80	82

   For each prime tex2html_wrap_inline$p$, let displaymath &delta#delta;(p) = S_4(&Gamma#Gamma;_0(p)) - S_p+3(&Gamma#Gamma;_0(1)). Then tex2html_wrap_inline$|&delta#delta;(p) - d_4(&Gamma#Gamma;_0(p))| &le#leq;2$ for each tex2html_wrap_inline$p<500$. Moreover, for every tex2html_wrap_inline$p&ne#neq;139$ we have that tex2html_wrap_inline$&delta#delta;(p)&ge#geq; d_4(&Gamma#Gamma;_0(p))$, but for tex2html_wrap_inline$p=139$, tex2html_wrap_inline$&delta#delta;(p)=23$ but tex2html_wrap_inline$d_4(&Gamma#Gamma;_0(p))=24$.