Space of modular forms of level 400, weight 4, and Character 400.c

• Label: 400.4.c
• Level: $400$
• Weight: $4$
• Character orbit: 400.c
• Rep. character: $\chi_{400}(49,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $13$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(400, [\chi])\).

	Total	New	Old
Modular forms	198	28	170
Cusp forms	162	26	136
Eisenstein series	36	2	34

Trace form

26 * q - 214 * q^9 + 68 * q^11 - 140 * q^19 + 80 * q^21 - 84 * q^29 - 448 * q^31 + 808 * q^39 + 176 * q^41 - 1490 * q^49 + 236 * q^51 - 1752 * q^59 - 524 * q^61 + 1600 * q^69 + 872 * q^71 + 1448 * q^79 + 1090 * q^81 + 1392 * q^89 + 1256 * q^91 - 9720 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(400, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.4.c.a	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					40.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+5\beta q^{3}+9\beta q^{7}-73 q^{9}+16 q^{11}+\cdots\)
400.4.c.b	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					200.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9 i q^{3}-26 i q^{7}-54 q^{9}+59 q^{11}+\cdots\)
400.4.c.c	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					10.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta q^{3}-2\beta q^{7}-37 q^{9}-12 q^{11}+\cdots\)
400.4.c.d	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					50.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+7 i q^{3}+34 i q^{7}-22 q^{9}-27 q^{11}+\cdots\)
400.4.c.e	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					25.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+7 i q^{3}-6 i q^{7}-22 q^{9}+43 q^{11}+\cdots\)
400.4.c.f	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					40.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta q^{3}-17\beta q^{7}-9 q^{9}-16 q^{11}+\cdots\)
400.4.c.g	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					200.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5 i q^{3}-2 i q^{7}+2 q^{9}-39 q^{11}+\cdots\)
400.4.c.h	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					40.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{3}-8\beta q^{7}+11 q^{9}-36 q^{11}+\cdots\)
400.4.c.i	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					8.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{3}+12\beta q^{7}+11 q^{9}+44 q^{11}+\cdots\)
400.4.c.j	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					20.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{3}+8\beta q^{7}+11 q^{9}+60 q^{11}+\cdots\)
400.4.c.k	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					5.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-3\beta q^{7}+23 q^{9}-32 q^{11}+\cdots\)
400.4.c.l	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					100.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-26 i q^{7}+26 q^{9}-45 q^{11}+\cdots\)
400.4.c.m	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None					200.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+6 i q^{7}+26 q^{9}+19 q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

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