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 - 33 * q^10 - 9 * q^11 - 89 * q^13 + 60 * q^14 - 217 * q^16 - 6 * q^17 - 39 * q^19 - 489 * q^20 + 138 * q^22 + 195 * q^23 - 626 * q^25 + 393 * q^26 + 240 * q^28 + 192 * q^29 - 837 * q^32 - 744 * q^35 - 396 * q^37 + 552 * q^38 + 954 * q^40 - 972 * q^41 - 647 * q^43 + 468 * q^46 - 169 * q^49 + 2328 * q^50 + 1394 * q^52 + 1422 * q^53 + 360 * q^55 + 1944 * q^56 + 2853 * q^58 - 1047 * q^59 + 350 * q^61 + 2730 * q^62 - 518 * q^64 + 759 * q^65 + 1455 * q^67 - 507 * q^68 - 2337 * q^71 - 3609 * q^74 - 1290 * q^76 + 1662 * q^77 - 6088 * q^79 - 9939 * q^80 - 1791 * q^82 - 2745 * q^85 - 1380 * q^88 - 3255 * q^89 + 1767 * q^91 + 5172 * q^92 - 594 * q^94 + 2334 * q^95 - 1917 * q^97 + 7989 * q^98

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.q.a	$117$	$4$	117.q	13.e	$6$	$2$	$1$	$6.903$	\(\Q(\sqrt{-3}) \)	None					13.4.e.b	$2$	$0$	\(-3\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\zeta_{6})q^{2}-5\zeta_{6}q^{4}+(-1+2\zeta_{6})q^{5}+\cdots\)
117.4.q.b	$117$	$4$	117.q	13.e	$6$	$2$	$1$	$6.903$	\(\Q(\sqrt{-3}) \)	None					39.4.j.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(18\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-8\zeta_{6}q^{4}+(-3+6\zeta_{6})q^{5}+(12-6\zeta_{6})q^{7}+\cdots\)
117.4.q.c	$117$	$4$	117.q	13.e	$6$	$2$	$1$	$6.903$	\(\Q(\sqrt{-3}) \)	None					13.4.e.a	$2$	$0$	\(6\)	\(0\)	\(0\)	\(39\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2+2\zeta_{6})q^{2}+4\zeta_{6}q^{4}+(8-2^{4}\zeta_{6})q^{5}+\cdots\)
117.4.q.d	$117$	$4$	117.q	13.e	$6$	$4$	$2$	$6.903$	\(\Q(\sqrt{-3}, \sqrt{-17})\)	None					39.4.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-66\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+9\beta _{2}q^{4}+(-3+6\beta _{2}+2\beta _{3})q^{5}+\cdots\)
117.4.q.e	$117$	$4$	117.q	13.e	$6$	$10$	$5$	$6.903$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					39.4.j.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(30\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(6\beta _{2}-\beta _{5})q^{4}+(-1+\cdots)q^{5}+\cdots\)
117.4.q.f	$117$	$4$	117.q	13.e	$6$	$12$	$6$	$6.903$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		117.4.q.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-18\)	$2^{4}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-3\beta _{3}-\beta _{8})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)

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

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