## Defining parameters

 Level: $$N$$ = $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$20$$ Newform subspaces: $$36$$ Sturm bound: $$5376$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(195))$$.

Total New Old
Modular forms 1536 979 557
Cusp forms 1153 851 302
Eisenstein series 383 128 255

## Trace form

 $$851q + 5q^{2} - 9q^{3} - 15q^{4} - q^{5} - 35q^{6} - 24q^{7} - 27q^{8} - 17q^{9} + O(q^{10})$$ $$851q + 5q^{2} - 9q^{3} - 15q^{4} - q^{5} - 35q^{6} - 24q^{7} - 27q^{8} - 17q^{9} - 61q^{10} - 4q^{11} - 59q^{12} - 57q^{13} - 24q^{14} - 27q^{15} - 119q^{16} - 22q^{17} - 55q^{18} - 92q^{19} - 57q^{20} - 80q^{21} - 116q^{22} - 24q^{23} - 63q^{24} - 49q^{25} - 83q^{26} - 93q^{27} - 136q^{28} - 26q^{29} - 41q^{30} - 88q^{31} - 47q^{32} - 20q^{33} - 62q^{34} - 16q^{35} + 33q^{36} + 22q^{37} + 68q^{38} + 31q^{39} - 111q^{40} - 62q^{41} - 24q^{42} - 116q^{43} - 44q^{44} - 7q^{45} - 240q^{46} - 64q^{47} + 41q^{48} - 129q^{49} - 85q^{50} - 74q^{51} - 67q^{52} - 22q^{53} + 37q^{54} - 4q^{55} + 48q^{56} - 12q^{57} - 22q^{58} + 44q^{59} + 173q^{60} - 138q^{61} + 120q^{62} + 112q^{63} + 149q^{64} + 117q^{65} + 140q^{66} + 84q^{67} + 310q^{68} + 168q^{69} + 144q^{70} + 136q^{71} + 297q^{72} + 94q^{73} + 154q^{74} + 113q^{75} + 140q^{76} + 96q^{77} + 281q^{78} - 16q^{79} + 63q^{80} + 79q^{81} - 58q^{82} - 12q^{83} + 88q^{84} - 112q^{85} - 52q^{86} - 34q^{87} - 228q^{88} - 138q^{89} - 43q^{90} - 264q^{91} - 168q^{92} - 104q^{93} - 296q^{94} - 156q^{95} - 211q^{96} - 194q^{97} - 155q^{98} - 148q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(195))$$

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
195.2.a $$\chi_{195}(1, \cdot)$$ 195.2.a.a 1 1
195.2.a.b 1
195.2.a.c 1
195.2.a.d 1
195.2.a.e 3
195.2.b $$\chi_{195}(181, \cdot)$$ 195.2.b.a 2 1
195.2.b.b 2
195.2.b.c 4
195.2.c $$\chi_{195}(79, \cdot)$$ 195.2.c.a 2 1
195.2.c.b 10
195.2.h $$\chi_{195}(64, \cdot)$$ 195.2.h.a 2 1
195.2.h.b 2
195.2.h.c 12
195.2.i $$\chi_{195}(16, \cdot)$$ 195.2.i.a 2 2
195.2.i.b 2
195.2.i.c 4
195.2.i.d 6
195.2.i.e 6
195.2.k $$\chi_{195}(112, \cdot)$$ 195.2.k.a 28 2
195.2.m $$\chi_{195}(53, \cdot)$$ 195.2.m.a 48 2
195.2.n $$\chi_{195}(44, \cdot)$$ 195.2.n.a 48 2
195.2.o $$\chi_{195}(86, \cdot)$$ 195.2.o.a 40 2
195.2.s $$\chi_{195}(38, \cdot)$$ 195.2.s.a 16 2
195.2.s.b 32
195.2.t $$\chi_{195}(73, \cdot)$$ 195.2.t.a 28 2
195.2.v $$\chi_{195}(4, \cdot)$$ 195.2.v.a 32 2
195.2.ba $$\chi_{195}(94, \cdot)$$ 195.2.ba.a 24 2
195.2.bb $$\chi_{195}(121, \cdot)$$ 195.2.bb.a 4 2
195.2.bb.b 8
195.2.bb.c 8
195.2.bd $$\chi_{195}(67, \cdot)$$ 195.2.bd.a 56 4
195.2.bf $$\chi_{195}(17, \cdot)$$ 195.2.bf.a 96 4
195.2.bg $$\chi_{195}(11, \cdot)$$ 195.2.bg.a 72 4
195.2.bh $$\chi_{195}(59, \cdot)$$ 195.2.bh.a 96 4
195.2.bl $$\chi_{195}(68, \cdot)$$ 195.2.bl.a 96 4
195.2.bm $$\chi_{195}(7, \cdot)$$ 195.2.bm.a 56 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(195))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(195)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 2}$$