## Defining parameters

 Level: $$N$$ = $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$69$$ Sturm bound: $$69984$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(810))$$.

Total New Old
Modular forms 18360 4224 14136
Cusp forms 16633 4224 12409
Eisenstein series 1727 0 1727

## Trace form

 $$4224q - 6q^{5} - 12q^{7} - 6q^{8} + O(q^{10})$$ $$4224q - 6q^{5} - 12q^{7} - 6q^{8} - 6q^{10} - 30q^{11} - 24q^{13} - 12q^{14} - 24q^{17} + 18q^{18} + 36q^{19} + 30q^{20} + 108q^{21} + 42q^{22} + 156q^{23} + 36q^{25} + 180q^{26} + 108q^{27} + 48q^{28} + 168q^{29} + 54q^{30} + 72q^{31} + 108q^{33} + 54q^{34} + 132q^{35} + 36q^{36} + 60q^{37} + 66q^{38} + 18q^{40} + 102q^{41} + 78q^{43} + 12q^{44} + 54q^{45} + 24q^{46} + 228q^{47} + 120q^{49} + 60q^{50} + 126q^{51} + 180q^{53} + 84q^{55} + 12q^{56} + 108q^{57} - 24q^{58} + 162q^{59} + 36q^{61} - 48q^{62} + 108q^{63} - 6q^{64} - 12q^{65} - 144q^{66} + 90q^{67} - 60q^{68} - 252q^{69} + 24q^{70} - 240q^{71} - 144q^{72} - 60q^{73} - 228q^{74} + 30q^{76} - 396q^{77} - 288q^{78} + 192q^{79} - 24q^{80} - 288q^{81} - 12q^{82} - 168q^{83} - 72q^{84} + 108q^{85} - 234q^{86} - 576q^{87} + 42q^{88} - 204q^{89} - 144q^{90} + 36q^{91} - 132q^{92} - 252q^{93} + 36q^{94} + 36q^{95} - 36q^{96} + 102q^{97} - 174q^{98} - 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(810))$$

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
810.2.a $$\chi_{810}(1, \cdot)$$ 810.2.a.a 1 1
810.2.a.b 1
810.2.a.c 1
810.2.a.d 1
810.2.a.e 1
810.2.a.f 1
810.2.a.g 1
810.2.a.h 1
810.2.a.i 2
810.2.a.j 2
810.2.a.k 2
810.2.a.l 2
810.2.c $$\chi_{810}(649, \cdot)$$ 810.2.c.a 2 1
810.2.c.b 2
810.2.c.c 2
810.2.c.d 2
810.2.c.e 4
810.2.c.f 4
810.2.c.g 8
810.2.e $$\chi_{810}(271, \cdot)$$ 810.2.e.a 2 2
810.2.e.b 2
810.2.e.c 2
810.2.e.d 2
810.2.e.e 2
810.2.e.f 2
810.2.e.g 2
810.2.e.h 2
810.2.e.i 2
810.2.e.j 2
810.2.e.k 2
810.2.e.l 2
810.2.e.m 4
810.2.e.n 4
810.2.f $$\chi_{810}(323, \cdot)$$ 810.2.f.a 8 2
810.2.f.b 8
810.2.f.c 16
810.2.f.d 16
810.2.i $$\chi_{810}(109, \cdot)$$ 810.2.i.a 4 2
810.2.i.b 4
810.2.i.c 4
810.2.i.d 4
810.2.i.e 4
810.2.i.f 4
810.2.i.g 8
810.2.i.h 8
810.2.i.i 8
810.2.k $$\chi_{810}(91, \cdot)$$ 810.2.k.a 6 6
810.2.k.b 12
810.2.k.c 12
810.2.k.d 18
810.2.k.e 24
810.2.m $$\chi_{810}(53, \cdot)$$ 810.2.m.a 8 4
810.2.m.b 8
810.2.m.c 8
810.2.m.d 8
810.2.m.e 8
810.2.m.f 8
810.2.m.g 8
810.2.m.h 8
810.2.m.i 16
810.2.m.j 16
810.2.p $$\chi_{810}(19, \cdot)$$ 810.2.p.a 108 6
810.2.q $$\chi_{810}(31, \cdot)$$ 810.2.q.a 144 18
810.2.q.b 162
810.2.q.c 162
810.2.q.d 180
810.2.s $$\chi_{810}(17, \cdot)$$ 810.2.s.a 216 12
810.2.v $$\chi_{810}(49, \cdot)$$ 810.2.v.a 972 18
810.2.w $$\chi_{810}(23, \cdot)$$ 810.2.w.a 1944 36

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(810))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(810)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(405))$$$$^{\oplus 2}$$