Space of modular forms of level 400, weight 8, and Character 400.c

• Label: 400.8.c
• Level: $400$
• Weight: $8$
• Character orbit: 400.c
• Rep. character: $\chi_{400}(49,\cdot)$
• Character field: $\Q$
• Dimension: $62$
• Newform subspaces: $21$
• Sturm bound: $480$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(480\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(400, [\chi])\).

	Total	New	Old
Modular forms	438	64	374
Cusp forms	402	62	340
Eisenstein series	36	2	34

Trace form

62 * q - 43738 * q^9 + 7988 * q^11 + 19428 * q^19 - 2032 * q^21 + 67604 * q^29 + 89280 * q^31 - 510776 * q^39 + 474984 * q^41 - 4531126 * q^49 + 5103980 * q^51 - 2778856 * q^59 - 2723700 * q^61 - 8414528 * q^69 - 20251896 * q^71 + 4908328 * q^79 + 31332838 * q^81 - 5474568 * q^89 + 1555304 * q^91 + 5997048 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.8.c.a	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					50.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+87 i q^{3}-1366 i q^{7}-5382 q^{9}+\cdots\)
400.8.c.b	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					8.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+42\beta q^{3}-228\beta q^{7}-4869 q^{9}+\cdots\)
400.8.c.c	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					200.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+69 i q^{3}+174 i q^{7}-2574 q^{9}+\cdots\)
400.8.c.d	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					50.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+57 i q^{3}+1174 i q^{7}-1062 q^{9}+\cdots\)
400.8.c.e	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					5.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+24\beta q^{3}-822\beta q^{7}-117 q^{9}+\cdots\)
400.8.c.f	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					8.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+22\beta q^{3}+612\beta q^{7}+251 q^{9}+\cdots\)
400.8.c.g	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					50.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+43 i q^{3}-974 i q^{7}+338 q^{9}+\cdots\)
400.8.c.h	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					40.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+18\beta q^{3}+388\beta q^{7}+891 q^{9}+\cdots\)
400.8.c.i	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					10.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+14\beta q^{3}-52\beta q^{7}+1403 q^{9}+\cdots\)
400.8.c.j	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					2.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+6\beta q^{3}-508\beta q^{7}+2043 q^{9}+\cdots\)
400.8.c.k	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					200.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9 i q^{3}+694 i q^{7}+2106 q^{9}+\cdots\)
400.8.c.l	$400$	$8$	400.c	5.b	$2$	$2$	$2$	$124.954$	\(\Q(\sqrt{-1}) \)	None					20.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta q^{3}-353\beta q^{7}+2151 q^{9}+\cdots\)
400.8.c.m	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{19})\)	None					5.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{3})q^{3}+(5\beta _{1}-7\beta _{3})q^{7}+(-2777+\cdots)q^{9}+\cdots\)
400.8.c.n	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{1129})\)	None					20.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{3}+(9\beta _{1}+83\beta _{2})q^{7}+(-2429+\cdots)q^{9}+\cdots\)
400.8.c.o	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{3889})\)	None					200.8.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}-\beta _{2})q^{3}+(-572\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
400.8.c.p	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{601})\)	None					40.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-19\beta _{1}+\beta _{2})q^{3}+(199\beta _{1}-7\beta _{2}+\cdots)q^{7}+\cdots\)
400.8.c.q	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{46})\)	None					40.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{3}+(-211\beta _{1}-17\beta _{2}+\cdots)q^{7}+\cdots\)
400.8.c.r	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{2521})\)	None					100.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+4\beta _{2})q^{3}+(6\beta _{1}-28\beta _{2})q^{7}+\cdots\)
400.8.c.s	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{649})\)	None					25.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+4\beta _{2})q^{3}+(-42\beta _{1}+60\beta _{2}+\cdots)q^{7}+\cdots\)
400.8.c.t	$400$	$8$	400.c	5.b	$2$	$4$	$4$	$124.954$	\(\Q(i, \sqrt{6})\)	None					40.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(9\beta _{1}+\beta _{2})q^{3}+(101\beta _{1}+63\beta _{2})q^{7}+\cdots\)
400.8.c.u	$400$	$8$	400.c	5.b	$2$	$6$	$6$	$124.954$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None			✓		200.8.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2^{5}\beta _{1}+\beta _{2})q^{3}+(-543\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(400, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)