## 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

 $$1133 q - 12 q^{2} - 18 q^{3} + 8 q^{4} + 12 q^{5} - 42 q^{6} - 80 q^{7} - 222 q^{8} - 114 q^{9} + O(q^{10})$$ $$1133 q - 12 q^{2} - 18 q^{3} + 8 q^{4} + 12 q^{5} - 42 q^{6} - 80 q^{7} - 222 q^{8} - 114 q^{9} + 12 q^{10} + 174 q^{11} + 288 q^{12} + 185 q^{13} + 204 q^{14} - 78 q^{15} - 160 q^{16} - 603 q^{17} - 456 q^{18} - 302 q^{19} - 558 q^{20} - 66 q^{21} + 306 q^{22} + 276 q^{23} + 270 q^{24} + 359 q^{25} + 1260 q^{26} + 816 q^{27} + 1852 q^{28} + 1947 q^{29} + 1032 q^{30} + 148 q^{31} - 942 q^{32} - 708 q^{33} - 1848 q^{34} - 3150 q^{35} - 2646 q^{36} - 1907 q^{37} - 5226 q^{38} - 2607 q^{39} - 6636 q^{40} - 3297 q^{41} - 924 q^{42} - 230 q^{43} - 396 q^{44} + 1566 q^{45} + 3558 q^{46} + 3360 q^{47} + 3762 q^{48} + 3846 q^{49} + 8244 q^{50} + 2214 q^{51} + 2930 q^{52} + 696 q^{53} - 2454 q^{54} - 1158 q^{55} + 2652 q^{56} + 2418 q^{57} + 888 q^{58} + 2070 q^{59} + 1548 q^{60} - 635 q^{61} + 8646 q^{62} + 5154 q^{63} + 14312 q^{64} + 7464 q^{65} + 8532 q^{66} + 5974 q^{67} + 6750 q^{68} - 2166 q^{69} - 3444 q^{70} - 10266 q^{71} - 9138 q^{72} - 9398 q^{73} - 17652 q^{74} - 7818 q^{75} - 21824 q^{76} - 14730 q^{77} - 16590 q^{78} - 11570 q^{79} - 29196 q^{80} - 5130 q^{81} - 17436 q^{82} - 6240 q^{83} - 7188 q^{84} - 1845 q^{85} - 4314 q^{86} + 1530 q^{87} + 11490 q^{88} + 5766 q^{89} + 13812 q^{90} + 10264 q^{91} + 25368 q^{92} + 12138 q^{93} + 30546 q^{94} + 19746 q^{95} - 4932 q^{96} + 11206 q^{97} - 3666 q^{98} - 14298 q^{99} + O(q^{100})$$

## 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 the 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

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