## Defining parameters

 Level: $$N$$ = $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$69$$ Sturm bound: $$37856$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(507))$$.

Total New Old
Modular forms 9920 7219 2701
Cusp forms 9009 6811 2198
Eisenstein series 911 408 503

## Trace form

 $$6811 q + 3 q^{2} - 65 q^{3} - 125 q^{4} + 6 q^{5} - 63 q^{6} - 132 q^{7} - 21 q^{8} - 69 q^{9} + O(q^{10})$$ $$6811 q + 3 q^{2} - 65 q^{3} - 125 q^{4} + 6 q^{5} - 63 q^{6} - 132 q^{7} - 21 q^{8} - 69 q^{9} - 174 q^{10} - 12 q^{11} - 111 q^{12} - 168 q^{13} - 24 q^{14} - 84 q^{15} - 181 q^{16} - 18 q^{17} - 111 q^{18} - 192 q^{19} - 90 q^{20} - 110 q^{21} - 216 q^{22} - 24 q^{23} - 123 q^{24} - 185 q^{25} - 60 q^{26} - 197 q^{27} - 244 q^{28} - 30 q^{29} - 96 q^{30} - 148 q^{31} - 57 q^{32} - 66 q^{33} - 174 q^{34} + q^{36} - 106 q^{37} + 60 q^{38} - 56 q^{39} - 258 q^{40} - 42 q^{41} - 78 q^{42} - 216 q^{43} - 36 q^{44} - 36 q^{45} - 300 q^{46} - 48 q^{47} - 107 q^{48} - 251 q^{49} - 135 q^{50} - 156 q^{51} - 294 q^{52} - 138 q^{53} - 159 q^{54} - 324 q^{55} - 240 q^{56} - 194 q^{57} - 342 q^{58} - 108 q^{59} - 228 q^{60} - 298 q^{61} - 168 q^{62} - 110 q^{63} - 353 q^{64} - 90 q^{65} - 114 q^{66} - 192 q^{67} - 54 q^{68} - 54 q^{69} - 252 q^{70} + 24 q^{71} - 39 q^{72} - 138 q^{73} - 42 q^{74} - 39 q^{75} - 216 q^{76} - 48 q^{77} - 18 q^{78} - 316 q^{79} - 90 q^{80} + 3 q^{81} - 210 q^{82} + 12 q^{83} - 110 q^{84} - 204 q^{85} - 60 q^{86} - 84 q^{87} - 360 q^{88} - 150 q^{89} - 108 q^{90} - 292 q^{91} - 168 q^{92} - 158 q^{93} - 420 q^{94} - 216 q^{95} - 231 q^{96} - 298 q^{97} - 141 q^{98} - 210 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(507))$$

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
507.2.a $$\chi_{507}(1, \cdot)$$ 507.2.a.a 1 1
507.2.a.b 1
507.2.a.c 1
507.2.a.d 2
507.2.a.e 2
507.2.a.f 2
507.2.a.g 2
507.2.a.h 2
507.2.a.i 3
507.2.a.j 3
507.2.a.k 3
507.2.a.l 3
507.2.b $$\chi_{507}(337, \cdot)$$ 507.2.b.a 2 1
507.2.b.b 2
507.2.b.c 2
507.2.b.d 4
507.2.b.e 4
507.2.b.f 6
507.2.b.g 6
507.2.e $$\chi_{507}(22, \cdot)$$ 507.2.e.a 2 2
507.2.e.b 2
507.2.e.c 2
507.2.e.d 4
507.2.e.e 4
507.2.e.f 4
507.2.e.g 4
507.2.e.h 4
507.2.e.i 6
507.2.e.j 6
507.2.e.k 6
507.2.e.l 6
507.2.f $$\chi_{507}(239, \cdot)$$ 507.2.f.a 4 2
507.2.f.b 4
507.2.f.c 4
507.2.f.d 8
507.2.f.e 8
507.2.f.f 8
507.2.f.g 48
507.2.j $$\chi_{507}(316, \cdot)$$ 507.2.j.a 2 2
507.2.j.b 2
507.2.j.c 2
507.2.j.d 4
507.2.j.e 4
507.2.j.f 8
507.2.j.g 8
507.2.j.h 12
507.2.j.i 12
507.2.k $$\chi_{507}(80, \cdot)$$ 507.2.k.a 4 4
507.2.k.b 4
507.2.k.c 4
507.2.k.d 8
507.2.k.e 8
507.2.k.f 8
507.2.k.g 8
507.2.k.h 8
507.2.k.i 8
507.2.k.j 8
507.2.k.k 96
507.2.m $$\chi_{507}(40, \cdot)$$ 507.2.m.a 180 12
507.2.m.b 204
507.2.p $$\chi_{507}(25, \cdot)$$ 507.2.p.a 168 12
507.2.p.b 192
507.2.q $$\chi_{507}(16, \cdot)$$ 507.2.q.a 360 24
507.2.q.b 384
507.2.s $$\chi_{507}(5, \cdot)$$ 507.2.s.a 1392 24
507.2.t $$\chi_{507}(4, \cdot)$$ 507.2.t.a 336 24
507.2.t.b 360
507.2.x $$\chi_{507}(2, \cdot)$$ 507.2.x.a 48 48
507.2.x.b 2784

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(507))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(507)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 2}$$