## Defining parameters

 Level: $$N$$ = $$512 = 2^{9}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$7$$ Newform subspaces: $$34$$ Sturm bound: $$32768$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(512))$$.

Total New Old
Modular forms 8576 4720 3856
Cusp forms 7809 4496 3313
Eisenstein series 767 224 543

## Trace form

 $$4496q - 64q^{2} - 48q^{3} - 64q^{4} - 64q^{5} - 64q^{6} - 48q^{7} - 64q^{8} - 80q^{9} + O(q^{10})$$ $$4496q - 64q^{2} - 48q^{3} - 64q^{4} - 64q^{5} - 64q^{6} - 48q^{7} - 64q^{8} - 80q^{9} - 64q^{10} - 48q^{11} - 64q^{12} - 64q^{13} - 64q^{14} - 48q^{15} - 64q^{16} - 96q^{17} - 64q^{18} - 48q^{19} - 64q^{20} - 64q^{21} - 64q^{22} - 48q^{23} - 64q^{24} - 80q^{25} - 64q^{26} - 48q^{27} - 64q^{28} - 64q^{29} - 64q^{30} - 48q^{31} - 64q^{32} - 112q^{33} - 64q^{34} - 48q^{35} - 64q^{36} - 64q^{37} - 64q^{38} - 48q^{39} - 64q^{40} - 80q^{41} - 64q^{42} - 48q^{43} - 64q^{44} - 64q^{45} - 64q^{46} - 48q^{47} - 64q^{48} - 96q^{49} - 64q^{50} - 48q^{51} - 64q^{52} - 64q^{53} - 64q^{54} - 48q^{55} - 64q^{56} - 80q^{57} - 64q^{58} - 48q^{59} - 64q^{60} - 64q^{61} - 64q^{62} - 64q^{63} - 64q^{64} - 128q^{65} - 64q^{66} - 48q^{67} - 64q^{68} - 64q^{69} - 64q^{70} - 48q^{71} - 64q^{72} - 80q^{73} - 64q^{74} - 48q^{75} - 64q^{76} - 64q^{77} - 64q^{78} - 48q^{79} - 64q^{80} - 96q^{81} - 64q^{82} - 48q^{83} - 64q^{84} - 64q^{85} - 64q^{86} - 48q^{87} - 64q^{88} - 80q^{89} - 64q^{90} - 48q^{91} - 64q^{92} - 112q^{93} - 64q^{94} - 48q^{95} - 64q^{96} - 112q^{97} - 64q^{98} - 48q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(512))$$

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
512.2.a $$\chi_{512}(1, \cdot)$$ 512.2.a.a 2 1
512.2.a.b 2
512.2.a.c 2
512.2.a.d 2
512.2.a.e 2
512.2.a.f 2
512.2.a.g 4
512.2.b $$\chi_{512}(257, \cdot)$$ 512.2.b.a 2 1
512.2.b.b 2
512.2.b.c 4
512.2.b.d 4
512.2.b.e 4
512.2.e $$\chi_{512}(129, \cdot)$$ 512.2.e.a 2 2
512.2.e.b 2
512.2.e.c 2
512.2.e.d 2
512.2.e.e 2
512.2.e.f 2
512.2.e.g 2
512.2.e.h 2
512.2.e.i 8
512.2.e.j 8
512.2.g $$\chi_{512}(65, \cdot)$$ 512.2.g.a 4 4
512.2.g.b 4
512.2.g.c 4
512.2.g.d 4
512.2.g.e 8
512.2.g.f 8
512.2.g.g 8
512.2.g.h 8
512.2.i $$\chi_{512}(33, \cdot)$$ 512.2.i.a 56 8
512.2.i.b 56
512.2.k $$\chi_{512}(17, \cdot)$$ 512.2.k.a 240 16
512.2.m $$\chi_{512}(9, \cdot)$$ None 0 32
512.2.o $$\chi_{512}(5, \cdot)$$ 512.2.o.a 4032 64

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(512))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(512)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(256))$$$$^{\oplus 2}$$