## Defining parameters

 Level: $$N$$ = $$510 = 2 \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newforms: $$60$$ Sturm bound: $$27648$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(510))$$.

Total New Old
Modular forms 7424 1569 5855
Cusp forms 6401 1569 4832
Eisenstein series 1023 0 1023

## Trace form

 $$1569q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 5q^{6} + 8q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$1569q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 5q^{6} + 8q^{7} + q^{8} + q^{9} + q^{10} + 44q^{11} + 5q^{12} + 46q^{13} + 40q^{14} + 29q^{15} + 17q^{16} + 41q^{17} + 49q^{18} + 52q^{19} + q^{20} + 88q^{21} + 60q^{22} + 40q^{23} + 21q^{24} + 89q^{25} + 30q^{26} + 5q^{27} + 8q^{28} + 62q^{29} + 29q^{30} + 128q^{31} + q^{32} + 60q^{33} - 15q^{34} + 40q^{35} + q^{36} + 38q^{37} - 28q^{38} - 42q^{39} - 23q^{40} + 42q^{41} - 120q^{42} + 12q^{43} - 4q^{44} - 47q^{45} - 72q^{46} - 48q^{47} - 11q^{48} - 87q^{49} - 15q^{50} - 139q^{51} - 18q^{52} - 26q^{53} - 107q^{54} + 12q^{55} - 8q^{56} - 124q^{57} - 2q^{58} + 44q^{59} - 19q^{60} + 62q^{61} - 16q^{62} + 8q^{63} + q^{64} + 22q^{65} - 52q^{66} + 36q^{67} - 39q^{68} - 8q^{69} - 184q^{70} - 184q^{71} + q^{72} - 342q^{73} - 186q^{74} - 187q^{75} - 44q^{76} - 288q^{77} - 138q^{78} - 288q^{79} - 87q^{80} - 47q^{81} - 294q^{82} - 236q^{83} - 72q^{84} - 591q^{85} - 196q^{86} - 346q^{87} - 132q^{88} - 358q^{89} - 47q^{90} - 528q^{91} - 88q^{92} - 320q^{93} - 272q^{94} - 412q^{95} - 11q^{96} - 382q^{97} - 183q^{98} - 148q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(510))$$

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
510.2.a $$\chi_{510}(1, \cdot)$$ 510.2.a.a 1 1
510.2.a.b 1
510.2.a.c 1
510.2.a.d 1
510.2.a.e 1
510.2.a.f 1
510.2.a.g 1
510.2.a.h 2
510.2.c $$\chi_{510}(271, \cdot)$$ 510.2.c.a 2 1
510.2.c.b 2
510.2.c.c 4
510.2.c.d 4
510.2.d $$\chi_{510}(409, \cdot)$$ 510.2.d.a 2 1
510.2.d.b 4
510.2.d.c 4
510.2.d.d 6
510.2.f $$\chi_{510}(169, \cdot)$$ 510.2.f.a 4 1
510.2.f.b 4
510.2.f.c 6
510.2.f.d 6
510.2.i $$\chi_{510}(353, \cdot)$$ 510.2.i.a 72 2
510.2.l $$\chi_{510}(137, \cdot)$$ 510.2.l.a 4 2
510.2.l.b 4
510.2.l.c 4
510.2.l.d 8
510.2.l.e 8
510.2.l.f 8
510.2.l.g 28
510.2.m $$\chi_{510}(259, \cdot)$$ 510.2.m.a 8 2
510.2.m.b 8
510.2.m.c 12
510.2.m.d 12
510.2.p $$\chi_{510}(361, \cdot)$$ 510.2.p.a 4 2
510.2.p.b 4
510.2.p.c 8
510.2.p.d 8
510.2.q $$\chi_{510}(203, \cdot)$$ 510.2.q.a 72 2
510.2.t $$\chi_{510}(47, \cdot)$$ 510.2.t.a 72 2
510.2.u $$\chi_{510}(121, \cdot)$$ 510.2.u.a 8 4
510.2.u.b 8
510.2.u.c 16
510.2.u.d 16
510.2.w $$\chi_{510}(257, \cdot)$$ 510.2.w.a 4 4
510.2.w.b 4
510.2.w.c 68
510.2.w.d 68
510.2.z $$\chi_{510}(53, \cdot)$$ 510.2.z.a 4 4
510.2.z.b 4
510.2.z.c 68
510.2.z.d 68
510.2.bb $$\chi_{510}(19, \cdot)$$ 510.2.bb.a 32 4
510.2.bb.b 32
510.2.bd $$\chi_{510}(7, \cdot)$$ 510.2.bd.a 64 8
510.2.bd.b 80
510.2.bf $$\chi_{510}(11, \cdot)$$ 510.2.bf.a 96 8
510.2.bf.b 96
510.2.bh $$\chi_{510}(29, \cdot)$$ 510.2.bh.a 144 8
510.2.bh.b 144
510.2.bi $$\chi_{510}(37, \cdot)$$ 510.2.bi.a 64 8
510.2.bi.b 80

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(510))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(510)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(85))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(102))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(170))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(255))$$$$^{\oplus 2}$$