Space of modular forms of level 13, weight 4, and Character 13.b

• Label: 13.4.b
• Level: $13$
• Weight: $4$
• Character orbit: 13.b
• Rep. character: $\chi_{13}(12,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $4$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
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:			\(4\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

	Total	New	Old
Modular forms	4	4	0
Cusp forms	2	2	0
Eisenstein series	2	2	0

Trace form

2 * q - 2 * q^3 - 2 * q^4 - 52 * q^9 + 54 * q^10 + 2 * q^12 + 52 * q^13 + 90 * q^14 - 142 * q^16 - 90 * q^17 - 288 * q^22 + 324 * q^23 + 88 * q^25 + 234 * q^26 + 106 * q^27 - 288 * q^29 - 54 * q^30 - 270 * q^35 + 52 * q^36 - 36 * q^38 - 52 * q^39 + 378 * q^40 - 90 * q^42 - 194 * q^43 + 142 * q^48 + 236 * q^49 + 90 * q^51 - 52 * q^52 - 828 * q^53 + 864 * q^55 + 630 * q^56 + 752 * q^61 - 1584 * q^62 - 866 * q^64 - 702 * q^65 + 288 * q^66 + 90 * q^68 - 324 * q^69 + 1818 * q^74 - 88 * q^75 + 1440 * q^77 - 234 * q^78 - 1660 * q^79 + 1298 * q^81 + 1152 * q^82 + 288 * q^87 - 2016 * q^88 - 1404 * q^90 - 1170 * q^91 - 324 * q^92 + 666 * q^94 + 108 * q^95

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
13.4.b.a	$13$	$4$	13.b	13.b	$2$	$2$	$2$	$0.767$	\(\Q(\sqrt{-1}) \)	None		✓	✓	✓	13.4.b.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-q^{3}-q^{4}-3\beta q^{5}-\beta q^{6}+\cdots\)