## Defining parameters

 Level: $$N$$ = $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$25$$ Sturm bound: $$3840$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(165))$$.

Total New Old
Modular forms 1120 703 417
Cusp forms 801 599 202
Eisenstein series 319 104 215

## Trace form

 $$599 q + 5 q^{2} - 7 q^{3} - 11 q^{4} - q^{5} - 39 q^{6} - 32 q^{7} - 31 q^{8} - 31 q^{9} + O(q^{10})$$ $$599 q + 5 q^{2} - 7 q^{3} - 11 q^{4} - q^{5} - 39 q^{6} - 32 q^{7} - 31 q^{8} - 31 q^{9} - 55 q^{10} - 5 q^{11} - 55 q^{12} - 22 q^{13} - 36 q^{14} - 32 q^{15} - 147 q^{16} - 46 q^{17} - 35 q^{18} - 68 q^{19} - 41 q^{20} - 72 q^{21} - 111 q^{22} - 16 q^{23} - 39 q^{24} - 71 q^{25} - 62 q^{26} + 23 q^{27} - 64 q^{28} - 6 q^{29} - 19 q^{30} - 68 q^{31} + 33 q^{32} + 17 q^{33} - 62 q^{34} - 32 q^{35} - 31 q^{36} - 42 q^{37} - 32 q^{38} - 50 q^{39} + 29 q^{40} - 98 q^{41} - 16 q^{42} - 44 q^{43} + 13 q^{44} - 76 q^{45} - 88 q^{46} + 12 q^{47} + 29 q^{48} + 31 q^{49} + 55 q^{50} - 38 q^{51} + 134 q^{52} + 54 q^{53} + 21 q^{54} + 65 q^{55} + 200 q^{56} + 108 q^{57} + 206 q^{58} + 128 q^{59} + 125 q^{60} + 66 q^{61} + 136 q^{62} + 68 q^{63} + 213 q^{64} + 78 q^{65} + 97 q^{66} + 24 q^{67} + 90 q^{68} + 34 q^{69} - 76 q^{70} - 32 q^{71} + 9 q^{72} - 146 q^{73} - 86 q^{74} - 82 q^{75} - 292 q^{76} - 92 q^{77} - 74 q^{78} - 200 q^{79} - 177 q^{80} - 51 q^{81} - 94 q^{82} - 200 q^{83} - 4 q^{84} - 146 q^{85} - 140 q^{86} - 114 q^{87} - 163 q^{88} - 58 q^{89} + 105 q^{90} - 88 q^{91} + 28 q^{92} - 18 q^{93} + 120 q^{94} + 22 q^{95} + 173 q^{96} + 74 q^{97} + 157 q^{98} + 145 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(165))$$

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
165.2.a $$\chi_{165}(1, \cdot)$$ 165.2.a.a 2 1
165.2.a.b 2
165.2.a.c 3
165.2.c $$\chi_{165}(34, \cdot)$$ 165.2.c.a 6 1
165.2.c.b 6
165.2.d $$\chi_{165}(164, \cdot)$$ 165.2.d.a 2 1
165.2.d.b 2
165.2.d.c 16
165.2.f $$\chi_{165}(131, \cdot)$$ 165.2.f.a 8 1
165.2.f.b 8
165.2.j $$\chi_{165}(43, \cdot)$$ 165.2.j.a 24 2
165.2.k $$\chi_{165}(23, \cdot)$$ 165.2.k.a 4 2
165.2.k.b 4
165.2.k.c 16
165.2.k.d 16
165.2.m $$\chi_{165}(16, \cdot)$$ 165.2.m.a 8 4
165.2.m.b 8
165.2.m.c 8
165.2.m.d 8
165.2.p $$\chi_{165}(41, \cdot)$$ 165.2.p.a 16 4
165.2.p.b 48
165.2.r $$\chi_{165}(29, \cdot)$$ 165.2.r.a 80 4
165.2.s $$\chi_{165}(4, \cdot)$$ 165.2.s.a 48 4
165.2.v $$\chi_{165}(38, \cdot)$$ 165.2.v.a 160 8
165.2.w $$\chi_{165}(7, \cdot)$$ 165.2.w.a 96 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(165))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(165)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 2}$$