Space of modular forms of level 13, weight 10, and Character 13.b

• Label: 13.10.b
• Level: $13$
• Weight: $10$
• Character orbit: 13.b
• Rep. character: $\chi_{13}(12,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $1$
• Sturm bound: $11$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	13.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(11\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(13, [\chi])\).

	Total	New	Old
Modular forms	12	12	0
Cusp forms	10	10	0
Eisenstein series	2	2	0

Trace form

\( 10 q - 2 q^{3} - 2562 q^{4} + 67304 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(13, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
13.10.b.a	$13$	$10$	13.b	13.b	$2$	$10$	$10$	$6.695$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2^{7}\cdot 3^{4}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(-2^{8}+\beta _{2})q^{4}+\cdots\)