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

• Label: 13.10.c
• Level: $13$
• Weight: $10$
• Character orbit: 13.c
• Rep. character: $\chi_{13}(3,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $18$
• Newform subspaces: $1$
• Sturm bound: $11$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	13.c (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
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	22	22	0
Cusp forms	18	18	0
Eisenstein series	4	4	0

Trace form

\( 18 q + 15 q^{2} + 161 q^{3} - 1793 q^{4} - 2280 q^{5} + 2118 q^{6} - 1939 q^{7} - 14478 q^{8} - 33654 q^{9} + O(q^{10}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
13.10.c.a	$13$	$10$	13.c	13.c	$3$	$18$	$9$	$6.695$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓	✓	✓	13.10.c.a	$2$	$0$	\(15\)	\(161\)	\(-2280\)	\(-1939\)	$2^{16}\cdot 3^{3}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2\beta _{2}-\beta _{3})q^{2}+(-18\beta _{2}-\beta _{6}+\cdots)q^{3}+\cdots\)