Space of modular forms of level 117 and weight 8

• Label: 117.8
• Level: 117
• Weight: 8
• Dimension: 2709
• Nonzero newspaces: 15
• Sturm bound: 8064
• Trace bound: 4

Defining parameters

Level:	\( N \)	=	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	=	\( 8 \)
Nonzero newspaces:			\( 15 \)
Sturm bound:			\(8064\)
Trace bound:			\(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(117))\).

	Total	New	Old
Modular forms	3624	2807	817
Cusp forms	3432	2709	723
Eisenstein series	192	98	94

Trace form

2709 * q + 12 * q^2 - 72 * q^3 - 120 * q^4 + 1122 * q^5 + 2442 * q^6 - 2276 * q^7 - 9894 * q^8 - 2004 * q^9 + 7842 * q^10 + 18960 * q^11 - 16128 * q^12 + 8112 * q^13 + 20892 * q^14 + 2352 * q^15 - 95792 * q^16 - 70137 * q^17 - 85776 * q^18 - 10850 * q^19 - 96498 * q^20 + 374424 * q^21 + 322950 * q^22 + 87546 * q^23 - 429426 * q^24 + 20769 * q^25 - 157608 * q^26 - 644592 * q^27 + 464652 * q^28 - 727959 * q^29 + 608712 * q^30 + 174396 * q^31 + 5421714 * q^32 + 1733880 * q^33 + 1871340 * q^34 - 2892060 * q^35 - 6766974 * q^36 - 1928841 * q^37 - 2124246 * q^38 - 361284 * q^39 + 93684 * q^40 + 4950795 * q^41 + 2204868 * q^42 + 718624 * q^43 + 1245348 * q^44 - 476244 * q^45 - 1821498 * q^46 + 982572 * q^47 + 5639202 * q^48 - 148862 * q^49 - 3806406 * q^50 - 3886992 * q^51 + 20723550 * q^52 - 830352 * q^53 - 11633082 * q^54 - 9928848 * q^55 - 8458692 * q^56 + 6143676 * q^57 - 3853650 * q^58 - 383940 * q^59 + 194508 * q^60 + 11397903 * q^61 + 45597918 * q^62 - 2914752 * q^63 - 22243896 * q^64 - 29878401 * q^65 - 33710652 * q^66 - 28259882 * q^67 + 43044642 * q^68 + 31113768 * q^69 + 73788036 * q^70 + 47354226 * q^71 + 49693038 * q^72 + 20321218 * q^73 - 62726334 * q^74 - 73486608 * q^75 - 56563592 * q^76 - 50483748 * q^77 - 86623698 * q^78 - 26721720 * q^79 - 1834956 * q^80 + 56965212 * q^81 - 18862974 * q^82 + 38199288 * q^83 + 39105948 * q^84 + 74798415 * q^85 + 87004146 * q^86 + 30576672 * q^87 + 206230242 * q^88 + 71291442 * q^89 - 57112428 * q^90 - 66347480 * q^91 - 286850040 * q^92 - 142251672 * q^93 - 180625998 * q^94 - 28035774 * q^95 + 84958236 * q^96 + 40815978 * q^97 + 139524690 * q^98 + 183775080 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(117))\)

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
117.8.a	\(\chi_{117}(1, \cdot)\)	117.8.a.a	1	1
		117.8.a.b	2
		117.8.a.c	2
		117.8.a.d	3
		117.8.a.e	4
		117.8.a.f	4
		117.8.a.g	5
		117.8.a.h	6
		117.8.a.i	8
117.8.b	\(\chi_{117}(64, \cdot)\)	117.8.b.a	4	1
		117.8.b.b	6
		117.8.b.c	8
		117.8.b.d	10
		117.8.b.e	12
117.8.e	\(\chi_{117}(40, \cdot)\)	n/a	168	2
117.8.f	\(\chi_{117}(61, \cdot)\)	n/a	192	2
117.8.g	\(\chi_{117}(55, \cdot)\)	117.8.g.a	2	2
		117.8.g.b	14
		117.8.g.c	16
		117.8.g.d	16
		117.8.g.e	32
117.8.h	\(\chi_{117}(16, \cdot)\)	n/a	192	2
117.8.i	\(\chi_{117}(8, \cdot)\)	117.8.i.a	68	2
117.8.l	\(\chi_{117}(4, \cdot)\)	n/a	192	2
117.8.q	\(\chi_{117}(10, \cdot)\)	117.8.q.a	2	2
		117.8.q.b	14
		117.8.q.c	16
		117.8.q.d	18
		117.8.q.e	28
117.8.r	\(\chi_{117}(43, \cdot)\)	n/a	192	2
117.8.t	\(\chi_{117}(25, \cdot)\)	n/a	192	2
117.8.x	\(\chi_{117}(2, \cdot)\)	n/a	384	4
117.8.z	\(\chi_{117}(5, \cdot)\)	n/a	384	4
117.8.ba	\(\chi_{117}(71, \cdot)\)	n/a	128	4
117.8.bc	\(\chi_{117}(20, \cdot)\)	n/a	384	4

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(117))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(117)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 2}\)