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 + 102 * q^10 + 38 * q^13 + 174 * q^14 + 154 * q^16 - 222 * q^17 + 432 * q^22 - 156 * q^23 - 226 * q^25 + 198 * q^26 + 720 * q^29 - 714 * q^35 + 12 * q^38 - 822 * q^40 + 350 * q^43 - 554 * q^49 + 292 * q^52 + 708 * q^53 - 2112 * q^55 - 2358 * q^56 - 1604 * q^61 - 1008 * q^62 + 2054 * q^64 - 570 * q^65 + 5694 * q^68 - 2658 * q^74 - 2040 * q^77 + 1180 * q^79 + 960 * q^82 - 1344 * q^88 + 390 * q^91 + 7884 * q^92 + 522 * q^94 - 1620 * q^95

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.4.b.a	$117$	$4$	117.b	13.b	$2$	$2$	$2$	$6.903$	\(\Q(\sqrt{-1}) \)	None					13.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{2}-q^{4}-3\beta q^{5}+5\beta q^{7}+7\beta q^{8}+\cdots\)
117.4.b.b	$117$	$4$	117.b	13.b	$2$	$2$	$2$	$6.903$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			117.4.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+8 q^{4}+\beta q^{7}+(-\beta+35)q^{13}+\cdots\)
117.4.b.c	$117$	$4$	117.b	13.b	$2$	$4$	$4$	$6.903$	4.0.8112.1	\(\Q(\sqrt{-39}) \)		✓			117.4.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(-8-\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{5}+\cdots\)
117.4.b.d	$117$	$4$	117.b	13.b	$2$	$4$	$4$	$6.903$	4.0.5054412.1	None					39.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}-6\beta _{1}q^{7}+\cdots\)
117.4.b.e	$117$	$4$	117.b	13.b	$2$	$4$	$4$	$6.903$	4.0.1362828.1	None					39.4.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-4+\beta _{3})q^{4}+(2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

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

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