## 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

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

## 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

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