Space of modular forms of level 117, weight 2, and Character 117.b

• Label: 117.2.b
• Level: $117$
• Weight: $2$
• Character orbit: 117.b
• Rep. character: $\chi_{117}(64,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $28$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$117 = 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	117.b (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$13$
Character field:			$\Q$
Newform subspaces:			$2$
Sturm bound:			$28$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(117, [\chi])$.

	Total	New	Old
Modular forms	18	8	10
Cusp forms	10	6	4
Eisenstein series	8	2	6

Trace form

6 * q - 10 * q^4 - 4 * q^10 - 2 * q^13 - 12 * q^14 + 26 * q^16 + 12 * q^17 - 16 * q^22 - 10 * q^25 + 12 * q^26 - 12 * q^29 + 12 * q^38 + 52 * q^40 - 24 * q^43 + 18 * q^49 - 50 * q^52 - 12 * q^53 + 32 * q^55 - 12 * q^56 - 4 * q^61 - 12 * q^62 - 74 * q^64 - 12 * q^68 + 24 * q^74 + 24 * q^77 - 16 * q^79 + 92 * q^82 + 64 * q^88 + 24 * q^91 + 8 * q^94

Decomposition of $S_{2}^{\mathrm{new}}(117, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.2.b.a	$117$	$2$	117.b	13.b	$2$	$2$	$2$	$0.934$	$\Q(\sqrt{-3})$	None					39.2.b.a	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta q^{2}-q^{4}-2\beta q^{7}-\beta q^{8}+2\beta q^{11}+\cdots$
117.2.b.b	$117$	$2$	117.b	13.b	$2$	$4$	$4$	$0.934$	4.0.8112.1	$\Q(\sqrt{-39})$		✓	✓		117.2.b.b	$4$	$0$	$0$	$0$	$0$	$0$	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q-\beta _{2}q^{2}+(-2-\beta _{3})q^{4}-\beta _{1}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(117, [\chi])$ into lower level spaces

$S_{2}^{\mathrm{old}}(117, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(39, [\chi])$$^{\oplus 2}$