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

\( 8 q + 2 q^{2} - 2 q^{4} + 8 q^{5} + 2 q^{7} - 12 q^{8} + O(q^{10}) \)

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

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

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