## Defining parameters

 Level: $$N$$ = $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$45$$ Sturm bound: $$11520$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(285))$$.

Total New Old
Modular forms 3168 2071 1097
Cusp forms 2593 1871 722
Eisenstein series 575 200 375

## Trace form

 $$1871 q + 5 q^{2} - 15 q^{3} - 27 q^{4} - q^{5} - 53 q^{6} - 28 q^{7} + 9 q^{8} - 19 q^{9} + O(q^{10})$$ $$1871 q + 5 q^{2} - 15 q^{3} - 27 q^{4} - q^{5} - 53 q^{6} - 28 q^{7} + 9 q^{8} - 19 q^{9} - 49 q^{10} + 20 q^{11} - 37 q^{12} - 66 q^{13} - 48 q^{14} - 42 q^{15} - 219 q^{16} - 22 q^{17} - 31 q^{18} - 105 q^{19} - 63 q^{20} - 88 q^{21} - 116 q^{22} - 12 q^{23} - 69 q^{24} - 91 q^{25} - 34 q^{26} - 39 q^{27} - 136 q^{28} - 38 q^{29} - 89 q^{30} - 148 q^{31} - 107 q^{32} - 122 q^{33} - 158 q^{34} - 64 q^{35} - 189 q^{36} - 86 q^{37} - 191 q^{38} - 134 q^{39} - 207 q^{40} - 50 q^{41} - 174 q^{42} - 144 q^{43} - 104 q^{44} - 55 q^{45} - 216 q^{46} - 40 q^{47} + 11 q^{48} - 37 q^{49} - 49 q^{50} + 40 q^{51} + 58 q^{52} + 74 q^{53} + 91 q^{54} - 34 q^{55} + 120 q^{56} + 95 q^{57} + 14 q^{58} + 68 q^{59} + 41 q^{60} - 162 q^{61} - 84 q^{62} + 38 q^{63} - 211 q^{64} - 90 q^{65} - 10 q^{66} - 280 q^{67} - 122 q^{68} - 84 q^{69} - 246 q^{70} - 56 q^{71} + 9 q^{72} - 306 q^{73} - 194 q^{74} - 57 q^{75} - 331 q^{76} - 228 q^{77} - 116 q^{78} - 220 q^{79} + 213 q^{80} + 17 q^{81} - 286 q^{82} + 24 q^{83} + 38 q^{84} + 32 q^{85} + 32 q^{86} + 8 q^{87} + 240 q^{88} - 6 q^{89} + 275 q^{90} - 44 q^{91} + 276 q^{92} + 98 q^{93} + 376 q^{94} + 177 q^{95} + 377 q^{96} + 166 q^{97} + 445 q^{98} + 272 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(285))$$

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
285.2.a $$\chi_{285}(1, \cdot)$$ 285.2.a.a 1 1
285.2.a.b 1
285.2.a.c 1
285.2.a.d 2
285.2.a.e 2
285.2.a.f 2
285.2.a.g 2
285.2.b $$\chi_{285}(284, \cdot)$$ 285.2.b.a 4 1
285.2.b.b 16
285.2.b.c 16
285.2.c $$\chi_{285}(229, \cdot)$$ 285.2.c.a 6 1
285.2.c.b 14
285.2.h $$\chi_{285}(56, \cdot)$$ 285.2.h.a 28 1
285.2.i $$\chi_{285}(106, \cdot)$$ 285.2.i.a 2 2
285.2.i.b 2
285.2.i.c 2
285.2.i.d 4
285.2.i.e 4
285.2.i.f 10
285.2.k $$\chi_{285}(77, \cdot)$$ 285.2.k.a 4 2
285.2.k.b 4
285.2.k.c 28
285.2.k.d 36
285.2.m $$\chi_{285}(37, \cdot)$$ 285.2.m.a 4 2
285.2.m.b 36
285.2.p $$\chi_{285}(221, \cdot)$$ 285.2.p.a 4 2
285.2.p.b 52
285.2.q $$\chi_{285}(164, \cdot)$$ 285.2.q.a 4 2
285.2.q.b 4
285.2.q.c 8
285.2.q.d 56
285.2.r $$\chi_{285}(49, \cdot)$$ 285.2.r.a 40 2
285.2.u $$\chi_{285}(16, \cdot)$$ 285.2.u.a 12 6
285.2.u.b 18
285.2.u.c 24
285.2.u.d 30
285.2.v $$\chi_{285}(68, \cdot)$$ 285.2.v.a 144 4
285.2.x $$\chi_{285}(88, \cdot)$$ 285.2.x.a 80 4
285.2.z $$\chi_{285}(41, \cdot)$$ 285.2.z.a 156 6
285.2.be $$\chi_{285}(4, \cdot)$$ 285.2.be.a 120 6
285.2.bf $$\chi_{285}(14, \cdot)$$ 285.2.bf.a 12 6
285.2.bf.b 12
285.2.bf.c 192
285.2.bh $$\chi_{285}(13, \cdot)$$ 285.2.bh.a 240 12
285.2.bi $$\chi_{285}(17, \cdot)$$ 285.2.bi.a 432 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(285))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(285)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 2}$$