Space of modular forms of level 410, weight 2, and Character 410.c

• Label: 410.2.c
• Level: $410$
• Weight: $2$
• Character orbit: 410.c
• Rep. character: $\chi_{410}(329,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $2$
• Sturm bound: $126$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 410 = 2 \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	410.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(126\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(410, [\chi])\).

	Total	New	Old
Modular forms	68	20	48
Cusp forms	60	20	40
Eisenstein series	8	0	8

Trace form

\( 20 q - 20 q^{4} + 4 q^{6} - 20 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(410, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
410.2.c.a	$410$	$2$	410.c	5.b	$2$	$6$	$6$	$3.274$	6.0.350464.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
410.2.c.b	$410$	$2$	410.c	5.b	$2$	$14$	$14$	$3.274$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{8})q^{3}-q^{4}+\beta _{6}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(410, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(410, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(205, [\chi])\)\(^{\oplus 2}\)