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

• Label: 400.2.s
• Level: $400$
• Weight: $2$
• Character orbit: 400.s
• Rep. character: $\chi_{400}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $68$
• Newform subspaces: $5$
• Sturm bound: $120$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.s (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 80 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(120\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	132	76	56
Cusp forms	108	68	40
Eisenstein series	24	8	16

Trace form

\( 68 q + 2 q^{2} + 4 q^{3} - 8 q^{4} - 4 q^{6} + 4 q^{7} + 8 q^{8} + 60 q^{9} + O(q^{10}) \)

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.s.a	$400$	$2$	400.s	80.s	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(4\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-i)q^{2}+2q^{3}-2iq^{4}+(2-2i)q^{6}+\cdots\)
400.2.s.b	$400$	$2$	400.s	80.s	$4$	$8$	$4$	$3.194$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\zeta_{24}-\zeta_{24}^{5})q^{2}+\zeta_{24}^{7}q^{3}-2q^{4}+\cdots\)
400.2.s.c	$400$	$2$	400.s	80.s	$4$	$16$	$8$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}+\beta _{5}q^{3}-\beta _{15}q^{4}+(-1-\beta _{11}+\cdots)q^{6}+\cdots\)
400.2.s.d	$400$	$2$	400.s	80.s	$4$	$18$	$9$	$3.194$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+\beta _{4}q^{3}-\beta _{14}q^{4}+(-1-\beta _{3}+\cdots)q^{6}+\cdots\)
400.2.s.e	$400$	$2$	400.s	80.s	$4$	$24$	$12$	$3.194$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)