Space of modular forms of level 117 and weight 4

• Label: 117.4
• Level: 117
• Weight: 4
• Dimension: 1133
• Nonzero newspaces: 15
• Newform subspaces: 38
• Sturm bound: 4032
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	=	\( 4 \)
Nonzero newspaces:			\( 15 \)
Newform subspaces:			\( 38 \)
Sturm bound:			\(4032\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(117))\).

	Total	New	Old
Modular forms	1608	1231	377
Cusp forms	1416	1133	283
Eisenstein series	192	98	94

Trace form

1133 * q - 12 * q^2 - 18 * q^3 + 8 * q^4 + 12 * q^5 - 42 * q^6 - 80 * q^7 - 222 * q^8 - 114 * q^9 + 12 * q^10 + 174 * q^11 + 288 * q^12 + 185 * q^13 + 204 * q^14 - 78 * q^15 - 160 * q^16 - 603 * q^17 - 456 * q^18 - 302 * q^19 - 558 * q^20 - 66 * q^21 + 306 * q^22 + 276 * q^23 + 270 * q^24 + 359 * q^25 + 1260 * q^26 + 816 * q^27 + 1852 * q^28 + 1947 * q^29 + 1032 * q^30 + 148 * q^31 - 942 * q^32 - 708 * q^33 - 1848 * q^34 - 3150 * q^35 - 2646 * q^36 - 1907 * q^37 - 5226 * q^38 - 2607 * q^39 - 6636 * q^40 - 3297 * q^41 - 924 * q^42 - 230 * q^43 - 396 * q^44 + 1566 * q^45 + 3558 * q^46 + 3360 * q^47 + 3762 * q^48 + 3846 * q^49 + 8244 * q^50 + 2214 * q^51 + 2930 * q^52 + 696 * q^53 - 2454 * q^54 - 1158 * q^55 + 2652 * q^56 + 2418 * q^57 + 888 * q^58 + 2070 * q^59 + 1548 * q^60 - 635 * q^61 + 8646 * q^62 + 5154 * q^63 + 14312 * q^64 + 7464 * q^65 + 8532 * q^66 + 5974 * q^67 + 6750 * q^68 - 2166 * q^69 - 3444 * q^70 - 10266 * q^71 - 9138 * q^72 - 9398 * q^73 - 17652 * q^74 - 7818 * q^75 - 21824 * q^76 - 14730 * q^77 - 16590 * q^78 - 11570 * q^79 - 29196 * q^80 - 5130 * q^81 - 17436 * q^82 - 6240 * q^83 - 7188 * q^84 - 1845 * q^85 - 4314 * q^86 + 1530 * q^87 + 11490 * q^88 + 5766 * q^89 + 13812 * q^90 + 10264 * q^91 + 25368 * q^92 + 12138 * q^93 + 30546 * q^94 + 19746 * q^95 - 4932 * q^96 + 11206 * q^97 - 3666 * q^98 - 14298 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(117))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label	\(\chi\)	Newforms	Dimension	\(\chi\) degree
117.4.a	\(\chi_{117}(1, \cdot)\)	117.4.a.a	1	1
		117.4.a.b	1
		117.4.a.c	2
		117.4.a.d	2
		117.4.a.e	2
		117.4.a.f	3
		117.4.a.g	4
117.4.b	\(\chi_{117}(64, \cdot)\)	117.4.b.a	2	1
		117.4.b.b	2
		117.4.b.c	4
		117.4.b.d	4
		117.4.b.e	4
117.4.e	\(\chi_{117}(40, \cdot)\)	117.4.e.a	36	2
		117.4.e.b	36
117.4.f	\(\chi_{117}(61, \cdot)\)	117.4.f.a	2	2
		117.4.f.b	78
117.4.g	\(\chi_{117}(55, \cdot)\)	117.4.g.a	2	2
		117.4.g.b	2
		117.4.g.c	2
		117.4.g.d	4
		117.4.g.e	8
		117.4.g.f	16
117.4.h	\(\chi_{117}(16, \cdot)\)	117.4.h.a	2	2
		117.4.h.b	78
117.4.i	\(\chi_{117}(8, \cdot)\)	117.4.i.a	28	2
117.4.l	\(\chi_{117}(4, \cdot)\)	117.4.l.a	80	2
117.4.q	\(\chi_{117}(10, \cdot)\)	117.4.q.a	2	2
		117.4.q.b	2
		117.4.q.c	2
		117.4.q.d	4
		117.4.q.e	10
		117.4.q.f	12
117.4.r	\(\chi_{117}(43, \cdot)\)	117.4.r.a	80	2
117.4.t	\(\chi_{117}(25, \cdot)\)	117.4.t.a	80	2
117.4.x	\(\chi_{117}(2, \cdot)\)	117.4.x.a	160	4
117.4.z	\(\chi_{117}(5, \cdot)\)	117.4.z.a	160	4
117.4.ba	\(\chi_{117}(71, \cdot)\)	117.4.ba.a	56	4
117.4.bc	\(\chi_{117}(20, \cdot)\)	117.4.bc.a	160	4

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(117))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(117)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 2}\)