## Defining parameters

 Level: $$N$$ = $$65 = 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$18$$ Sturm bound: $$672$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(65))$$.

Total New Old
Modular forms 216 185 31
Cusp forms 121 117 4
Eisenstein series 95 68 27

## Trace form

 $$117q - 15q^{2} - 16q^{3} - 19q^{4} - 19q^{5} - 48q^{6} - 16q^{7} - 9q^{8} - 9q^{9} + O(q^{10})$$ $$117q - 15q^{2} - 16q^{3} - 19q^{4} - 19q^{5} - 48q^{6} - 16q^{7} - 9q^{8} - 9q^{9} - 6q^{10} - 36q^{11} + 12q^{12} - q^{13} - 24q^{14} - 10q^{15} - 27q^{16} - 12q^{17} + 15q^{18} + 8q^{19} + 8q^{20} - 16q^{21} + 12q^{22} - 12q^{23} + 48q^{24} - 13q^{25} - 15q^{26} + 8q^{27} + 32q^{28} + 12q^{29} + 42q^{30} - 16q^{31} + 57q^{32} + 24q^{33} + 30q^{34} + 16q^{35} + 41q^{36} - 16q^{37} + 12q^{38} + 21q^{40} + 12q^{41} + 72q^{42} + 36q^{43} + 60q^{44} + 44q^{45} + 60q^{46} + 24q^{47} - 16q^{48} + 47q^{49} + 60q^{50} - 21q^{52} - 30q^{53} - 72q^{54} - 36q^{55} - 72q^{56} - 64q^{57} - 60q^{58} - 60q^{59} - 172q^{60} - 32q^{61} - 84q^{62} - 88q^{63} - 97q^{64} - 46q^{65} - 144q^{66} - 64q^{67} - 72q^{68} - 120q^{69} - 48q^{70} - 36q^{71} - 123q^{72} - 22q^{73} + 12q^{74} - 30q^{75} - 40q^{76} + 36q^{78} - 32q^{79} + 62q^{80} + 3q^{81} + 72q^{82} + 60q^{83} + 56q^{84} + 57q^{85} + 84q^{86} + 108q^{87} + 156q^{88} + 90q^{89} + 165q^{90} + 56q^{91} + 96q^{92} + 128q^{93} + 132q^{94} + 70q^{95} + 204q^{96} + 50q^{97} + 105q^{98} + 132q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$

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
65.2.a $$\chi_{65}(1, \cdot)$$ 65.2.a.a 1 1
65.2.a.b 2
65.2.a.c 2
65.2.b $$\chi_{65}(14, \cdot)$$ 65.2.b.a 6 1
65.2.c $$\chi_{65}(51, \cdot)$$ 65.2.c.a 6 1
65.2.d $$\chi_{65}(64, \cdot)$$ 65.2.d.a 2 1
65.2.d.b 2
65.2.e $$\chi_{65}(16, \cdot)$$ 65.2.e.a 4 2
65.2.e.b 4
65.2.f $$\chi_{65}(18, \cdot)$$ 65.2.f.a 2 2
65.2.f.b 8
65.2.k $$\chi_{65}(8, \cdot)$$ 65.2.k.a 2 2
65.2.k.b 8
65.2.l $$\chi_{65}(4, \cdot)$$ 65.2.l.a 8 2
65.2.m $$\chi_{65}(36, \cdot)$$ 65.2.m.a 8 2
65.2.n $$\chi_{65}(9, \cdot)$$ 65.2.n.a 12 2
65.2.o $$\chi_{65}(2, \cdot)$$ 65.2.o.a 20 4
65.2.t $$\chi_{65}(7, \cdot)$$ 65.2.t.a 20 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(65))$$ into lower level spaces

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