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

\( 8 q - 4 q^{9} - 4 q^{11} + 12 q^{19} - 16 q^{21} + 8 q^{29} - 40 q^{39} - 12 q^{41} + 12 q^{51} + 32 q^{59} + 24 q^{61} + 24 q^{71} + 8 q^{79} + 16 q^{81} + 12 q^{89} - 24 q^{91}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.2.c.a	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None					200.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+2 i q^{7}-6 q^{9}-q^{11}+\cdots\)
400.2.c.b	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None					20.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+\beta q^{7}-q^{9}+\beta q^{13}+3\beta q^{17}+\cdots\)
400.2.c.c	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None					50.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-2 i q^{7}+2 q^{9}+3 q^{11}+\cdots\)
400.2.c.d	$400$	$2$	400.c	5.b	$2$	$2$	$2$	$3.194$	\(\Q(\sqrt{-1}) \)	None					40.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta q^{7}+3 q^{9}-4 q^{11}+\beta q^{13}+\cdots\)

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

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