Space of modular forms of level 13, weight 2, and Character 13.e

• Label: 13.2.e
• Level: $13$
• Weight: $2$
• Character orbit: 13.e
• Rep. character: $\chi_{13}(4,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $2$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(2\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(13, [\chi])\).

	Total	New	Old
Modular forms	6	6	0
Cusp forms	2	2	0
Eisenstein series	4	4	0

Trace form

2 * q - 3 * q^2 - 2 * q^3 + q^4 + 6 * q^6 - q^9 - 3 * q^10 - 4 * q^12 - 5 * q^13 + 6 * q^15 + 5 * q^16 + 3 * q^17 - 6 * q^19 + 3 * q^20 + 6 * q^23 - 6 * q^24 + 4 * q^25 + 3 * q^26 - 8 * q^27 - 3 * q^29 - 6 * q^30 - 9 * q^32 + q^36 + 15 * q^37 + 12 * q^38 + 14 * q^39 + 6 * q^40 - 9 * q^41 - 8 * q^43 - 3 * q^45 - 18 * q^46 + 10 * q^48 - 7 * q^49 - 6 * q^50 - 12 * q^51 + 2 * q^52 - 6 * q^53 + 12 * q^54 + 9 * q^58 + 12 * q^59 - q^61 + 6 * q^62 - 2 * q^64 - 9 * q^65 + 6 * q^67 - 3 * q^68 + 12 * q^69 + 6 * q^71 + 3 * q^72 - 15 * q^74 - 4 * q^75 - 6 * q^76 - 24 * q^78 + 8 * q^79 - 15 * q^80 + 11 * q^81 + 9 * q^82 + 9 * q^85 - 6 * q^87 - 12 * q^89 + 6 * q^90 + 12 * q^92 - 12 * q^93 - 6 * q^94 + 6 * q^95 + 12 * q^97 + 21 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(13, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
13.2.e.a	$13$	$2$	13.e	13.e	$6$	$2$	$1$	$0.104$	\(\Q(\sqrt{-3}) \)	None		✓	✓	✓	13.2.e.a	$2$	$0$	\(-3\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)