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

\( 26 q - 214 q^{9} + O(q^{10}) \)

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	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		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+5iq^{3}+9iq^{7}-73q^{9}+2^{4}q^{11}+\cdots\)
400.4.c.b	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}-26iq^{7}-54q^{9}+59q^{11}+\cdots\)
400.4.c.c	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}-2iq^{7}-37q^{9}-12q^{11}+\cdots\)
400.4.c.d	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+7iq^{3}+34iq^{7}-22q^{9}-3^{3}q^{11}+\cdots\)
400.4.c.e	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+7iq^{3}-6iq^{7}-22q^{9}+43q^{11}+\cdots\)
400.4.c.f	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}-17iq^{7}-9q^{9}-2^{4}q^{11}+\cdots\)
400.4.c.g	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5iq^{3}-2iq^{7}+2q^{9}-39q^{11}+\cdots\)
400.4.c.h	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}-8iq^{7}+11q^{9}-6^{2}q^{11}+\cdots\)
400.4.c.i	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}+12iq^{7}+11q^{9}+44q^{11}+\cdots\)
400.4.c.j	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{3}+8iq^{7}+11q^{9}+60q^{11}+\cdots\)
400.4.c.k	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-3iq^{7}+23q^{9}-2^{5}q^{11}+\cdots\)
400.4.c.l	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-26iq^{7}+26q^{9}-45q^{11}+\cdots\)
400.4.c.m	$400$	$4$	400.c	5.b	$2$	$2$	$2$	$23.601$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+6iq^{7}+26q^{9}+19q^{11}+\cdots\)

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

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