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

• Label: 117.4.b
• Level: $117$
• Weight: $4$
• Character orbit: 117.b
• Rep. character: $\chi_{117}(64,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $5$
• Sturm bound: $56$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(56\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	46	18	28
Cusp forms	38	16	22
Eisenstein series	8	2	6

Trace form

\( 16 q - 58 q^{4} + O(q^{10}) \)

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.b.a	$117$	$4$	117.b	13.b	$2$	$2$	$2$	$6.903$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}-3iq^{5}+5iq^{7}+7iq^{8}+\cdots\)
117.4.b.b	$117$	$4$	117.b	13.b	$2$	$2$	$2$	$6.903$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+8q^{4}+\zeta_{6}q^{7}+(35-\zeta_{6})q^{13}+2^{6}q^{16}+\cdots\)
117.4.b.c	$117$	$4$	117.b	13.b	$2$	$4$	$4$	$6.903$	4.0.8112.1	\(\Q(\sqrt{-39}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(-8-\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{5}+\cdots\)
117.4.b.d	$117$	$4$	117.b	13.b	$2$	$4$	$4$	$6.903$	4.0.5054412.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}-6\beta _{1}q^{7}+\cdots\)
117.4.b.e	$117$	$4$	117.b	13.b	$2$	$4$	$4$	$6.903$	4.0.1362828.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-4+\beta _{3})q^{4}+(2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

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

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