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

• Label: 117.2.g
• Level: $117$
• Weight: $2$
• Character orbit: 117.g
• Rep. character: $\chi_{117}(55,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $28$
• Trace bound: $2$

Defining parameters

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

Dimensions

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

	Total	New	Old
Modular forms	36	12	24
Cusp forms	20	8	12
Eisenstein series	16	4	12

Trace form

8 * q + 2 * q^2 - 2 * q^4 + 8 * q^5 + 2 * q^7 - 12 * q^8 - 6 * q^10 - 6 * q^11 + 16 * q^14 - 6 * q^16 - 6 * q^17 - 8 * q^19 + 2 * q^20 + 4 * q^22 - 2 * q^23 - 12 * q^25 - 30 * q^26 + 16 * q^28 - 4 * q^31 + 14 * q^32 + 4 * q^34 - 6 * q^35 - 2 * q^37 + 40 * q^38 - 4 * q^40 + 10 * q^41 + 2 * q^43 + 16 * q^44 + 4 * q^46 - 12 * q^47 + 10 * q^49 - 28 * q^50 - 16 * q^52 - 4 * q^53 - 8 * q^55 - 28 * q^56 + 26 * q^58 - 14 * q^59 - 4 * q^61 - 4 * q^62 + 12 * q^64 - 4 * q^65 - 14 * q^67 + 18 * q^68 - 8 * q^70 + 34 * q^71 + 32 * q^73 - 2 * q^74 + 12 * q^76 - 4 * q^77 - 4 * q^79 + 22 * q^80 - 18 * q^82 + 48 * q^83 - 14 * q^85 + 12 * q^88 + 4 * q^89 + 38 * q^91 - 32 * q^92 + 28 * q^94 - 20 * q^95 + 10 * q^97 + 22 * q^98

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.g.a	$117$	$2$	117.g	13.c	$3$	$2$	$1$	$0.934$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			117.2.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2\zeta_{6}q^{4}+\zeta_{6}q^{7}+(4-3\zeta_{6})q^{13}+\cdots\)
117.2.g.b	$117$	$2$	117.g	13.c	$3$	$2$	$1$	$0.934$	\(\Q(\sqrt{-3}) \)	None					39.2.e.a	$2$	$0$	\(1\)	\(0\)	\(2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+q^{5}-2\zeta_{6}q^{7}+\cdots\)
117.2.g.c	$117$	$2$	117.g	13.c	$3$	$4$	$2$	$0.934$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None			✓		39.2.e.b	$2$	$0$	\(1\)	\(0\)	\(6\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-2+\beta _{1}+2\beta _{2}+\beta _{3})q^{4}+\cdots\)

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

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