## Defining parameters

 Level: $$N$$ = $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$16$$ Sturm bound: $$2592$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(135))$$.

Total New Old
Modular forms 768 518 250
Cusp forms 529 422 107
Eisenstein series 239 96 143

## Trace form

 $$422q - 4q^{2} - 12q^{3} - 12q^{4} - 13q^{5} - 48q^{6} - 14q^{7} - 36q^{8} - 24q^{9} + O(q^{10})$$ $$422q - 4q^{2} - 12q^{3} - 12q^{4} - 13q^{5} - 48q^{6} - 14q^{7} - 36q^{8} - 24q^{9} - 33q^{10} - 46q^{11} - 48q^{12} - 34q^{13} - 66q^{14} - 33q^{15} - 80q^{16} - 46q^{17} - 42q^{18} - 16q^{19} - 45q^{20} - 24q^{21} - 58q^{22} - 12q^{23} + 12q^{24} - 37q^{25} - 4q^{26} - 6q^{27} - 60q^{28} - 32q^{29} - 15q^{30} - 70q^{31} + 16q^{32} - 12q^{33} - 46q^{34} + 8q^{35} - 12q^{36} - 36q^{37} - 16q^{38} - 30q^{39} + 31q^{40} + 10q^{41} + 36q^{42} + 14q^{43} + 166q^{44} + 45q^{45} - 10q^{46} + 98q^{47} + 126q^{48} + 152q^{50} + 36q^{51} + 30q^{52} + 80q^{53} + 156q^{54} - 60q^{55} + 66q^{56} + 18q^{57} - 70q^{58} - 28q^{59} + 48q^{60} - 126q^{61} - 84q^{62} - 42q^{63} - 84q^{64} - 87q^{65} - 30q^{66} - 56q^{67} - 62q^{68} - 42q^{69} - 45q^{70} - 158q^{71} + 36q^{72} - 34q^{73} - 14q^{74} - 21q^{75} + 28q^{76} + 54q^{77} + 96q^{78} + 94q^{79} + 150q^{80} - 24q^{81} + 120q^{82} + 78q^{83} + 192q^{84} + 63q^{85} + 170q^{86} + 78q^{87} + 168q^{88} + 186q^{89} + 186q^{90} + 38q^{91} + 402q^{92} + 198q^{93} + 146q^{94} + 165q^{95} + 168q^{96} + 60q^{97} + 370q^{98} + 150q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$

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
135.2.a $$\chi_{135}(1, \cdot)$$ 135.2.a.a 1 1
135.2.a.b 1
135.2.a.c 2
135.2.a.d 2
135.2.b $$\chi_{135}(109, \cdot)$$ 135.2.b.a 4 1
135.2.b.b 4
135.2.e $$\chi_{135}(46, \cdot)$$ 135.2.e.a 2 2
135.2.e.b 6
135.2.f $$\chi_{135}(53, \cdot)$$ 135.2.f.a 8 2
135.2.f.b 8
135.2.j $$\chi_{135}(19, \cdot)$$ 135.2.j.a 8 2
135.2.k $$\chi_{135}(16, \cdot)$$ 135.2.k.a 30 6
135.2.k.b 42
135.2.m $$\chi_{135}(8, \cdot)$$ 135.2.m.a 16 4
135.2.p $$\chi_{135}(4, \cdot)$$ 135.2.p.a 96 6
135.2.q $$\chi_{135}(2, \cdot)$$ 135.2.q.a 192 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(135))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(135)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$