Space of modular forms of level 3, weight 13, and Character 3.b

• Label: 3.13.b
• Level: $3$
• Weight: $13$
• Character orbit: 3.b
• Rep. character: $\chi_{3}(2,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $4$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 3 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	3.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(4\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(3, [\chi])\).

	Total	New	Old
Modular forms	5	5	0
Cusp forms	3	3	0
Eisenstein series	2	2	0

Trace form

\( 3 q - 621 q^{3} - 4560 q^{4} + 50544 q^{6} - 73002 q^{7} + 1291059 q^{9} + O(q^{10}) \)

Decomposition of \(S_{13}^{\mathrm{new}}(3, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3.13.b.a	$3$	$13$	3.b	3.b	$2$	$1$	$1$	$2.742$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(729\)	\(0\)	\(-153502\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{6}q^{3}+2^{12}q^{4}-153502q^{7}+3^{12}q^{9}+\cdots\)
3.13.b.b	$3$	$13$	3.b	3.b	$2$	$2$	$2$	$2.742$	\(\Q(\sqrt{-26}) \)	None		$2$	$0$	\(0\)	\(-1350\)	\(0\)	\(80500\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}+(-675-3\beta )q^{3}-4328q^{4}+\cdots\)