Space of modular forms of level 400, weight 2, and Character 400.c

• Label: 400.2.c
• Level: $400$
• Weight: $2$
• Character orbit: 400.c
• Rep. character: $\chi_{400}(49,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(400, [\chi])\).

	Total	New	Old
Modular forms	78	10	68
Cusp forms	42	8	34
Eisenstein series	36	2	34

Trace form

\( 8 q - 4 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(400, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.2.c.a	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3iq^{3}+2iq^{7}-6q^{9}-q^{11}+4iq^{13}+\cdots\)
400.2.c.b	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}+iq^{13}+3iq^{17}+\cdots\)
400.2.c.c	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-2iq^{7}+2q^{9}+3q^{11}+4iq^{13}+\cdots\)
400.2.c.d	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2iq^{7}+3q^{9}-4q^{11}+iq^{13}+iq^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

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