Space of modular forms of level 117, weight 4, and Character 117.q

• Label: 117.4.q
• Level: $117$
• Weight: $4$
• Character orbit: 117.q
• Rep. character: $\chi_{117}(10,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $32$
• Newform subspaces: $6$
• Sturm bound: $56$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.q (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(56\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(117, [\chi])\).

	Total	New	Old
Modular forms	92	36	56
Cusp forms	76	32	44
Eisenstein series	16	4	12

Trace form

\( 32 q + 3 q^{2} + 55 q^{4} - 21 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(117, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.4.q.a	$117$	$4$	117.q	13.e	$6$	$2$	$1$	$6.903$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-3\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}-5\zeta_{6}q^{4}+(-1+2\zeta_{6})q^{5}+\cdots\)
117.4.q.b	$117$	$4$	117.q	13.e	$6$	$2$	$1$	$6.903$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(18\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-8\zeta_{6}q^{4}+(-3+6\zeta_{6})q^{5}+(12-6\zeta_{6})q^{7}+\cdots\)
117.4.q.c	$117$	$4$	117.q	13.e	$6$	$2$	$1$	$6.903$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(6\)	\(0\)	\(0\)	\(39\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2+2\zeta_{6})q^{2}+4\zeta_{6}q^{4}+(8-2^{4}\zeta_{6})q^{5}+\cdots\)
117.4.q.d	$117$	$4$	117.q	13.e	$6$	$4$	$2$	$6.903$	\(\Q(\sqrt{-3}, \sqrt{-17})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-66\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+9\beta _{2}q^{4}+(-3+6\beta _{2}+2\beta _{3})q^{5}+\cdots\)
117.4.q.e	$117$	$4$	117.q	13.e	$6$	$10$	$5$	$6.903$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(30\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(6\beta _{2}-\beta _{5})q^{4}+(-1+\cdots)q^{5}+\cdots\)
117.4.q.f	$117$	$4$	117.q	13.e	$6$	$12$	$6$	$6.903$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-18\)	$2^{4}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-3\beta _{3}-\beta _{8})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(117, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(117, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)