## Defining parameters

 Level: $$N$$ = $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$20$$ Newform subspaces: $$100$$ Sturm bound: $$76800$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(800))$$.

Total New Old
Modular forms 20096 10159 9937
Cusp forms 18305 9749 8556
Eisenstein series 1791 410 1381

## Trace form

 $$9749q - 52q^{2} - 40q^{3} - 52q^{4} - 64q^{5} - 84q^{6} - 40q^{7} - 52q^{8} - 79q^{9} + O(q^{10})$$ $$9749q - 52q^{2} - 40q^{3} - 52q^{4} - 64q^{5} - 84q^{6} - 40q^{7} - 52q^{8} - 79q^{9} - 64q^{10} - 64q^{11} - 36q^{12} - 46q^{13} - 36q^{14} - 48q^{15} - 64q^{16} - 22q^{17} - 32q^{18} - 32q^{19} - 64q^{20} - 84q^{21} - 40q^{22} - 32q^{23} - 64q^{24} - 96q^{25} - 184q^{26} + 8q^{27} - 72q^{28} - 46q^{29} - 64q^{30} - 12q^{31} - 72q^{32} - 64q^{33} - 64q^{34} - 24q^{35} - 112q^{36} + 26q^{37} - 44q^{38} + 40q^{39} - 64q^{40} - 50q^{41} - 32q^{42} + 32q^{43} - 12q^{44} - 24q^{45} - 52q^{46} + 12q^{47} + 33q^{49} - 64q^{50} - 96q^{51} - 44q^{52} - 6q^{53} - 40q^{54} - 28q^{55} - 72q^{56} - 124q^{57} - 56q^{58} - 64q^{59} - 128q^{60} - 62q^{61} - 72q^{62} - 116q^{63} - 184q^{64} - 192q^{65} - 364q^{66} - 128q^{67} - 256q^{68} - 212q^{69} - 208q^{70} - 160q^{71} - 508q^{72} - 210q^{73} - 388q^{74} - 104q^{75} - 420q^{76} - 228q^{77} - 620q^{78} - 156q^{79} - 224q^{80} - 287q^{81} - 372q^{82} - 80q^{83} - 640q^{84} - 160q^{85} - 360q^{86} - 72q^{87} - 400q^{88} - 194q^{89} - 304q^{90} - 80q^{91} - 376q^{92} - 256q^{93} - 264q^{94} - 52q^{95} - 336q^{96} - 174q^{97} - 168q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(800))$$

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
800.2.a $$\chi_{800}(1, \cdot)$$ 800.2.a.a 1 1
800.2.a.b 1
800.2.a.c 1
800.2.a.d 1
800.2.a.e 1
800.2.a.f 1
800.2.a.g 1
800.2.a.h 1
800.2.a.i 1
800.2.a.j 2
800.2.a.k 2
800.2.a.l 2
800.2.a.m 2
800.2.a.n 2
800.2.c $$\chi_{800}(449, \cdot)$$ 800.2.c.a 2 1
800.2.c.b 2
800.2.c.c 2
800.2.c.d 2
800.2.c.e 2
800.2.c.f 4
800.2.c.g 4
800.2.d $$\chi_{800}(401, \cdot)$$ 800.2.d.a 2 1
800.2.d.b 2
800.2.d.c 2
800.2.d.d 2
800.2.d.e 4
800.2.d.f 4
800.2.f $$\chi_{800}(49, \cdot)$$ 800.2.f.a 2 1
800.2.f.b 2
800.2.f.c 4
800.2.f.d 4
800.2.f.e 4
800.2.j $$\chi_{800}(407, \cdot)$$ None 0 2
800.2.l $$\chi_{800}(201, \cdot)$$ None 0 2
800.2.n $$\chi_{800}(543, \cdot)$$ 800.2.n.a 2 2
800.2.n.b 2
800.2.n.c 2
800.2.n.d 2
800.2.n.e 2
800.2.n.f 2
800.2.n.g 2
800.2.n.h 2
800.2.n.i 2
800.2.n.j 2
800.2.n.k 4
800.2.n.l 4
800.2.n.m 4
800.2.n.n 4
800.2.o $$\chi_{800}(143, \cdot)$$ 800.2.o.a 2 2
800.2.o.b 2
800.2.o.c 2
800.2.o.d 2
800.2.o.e 4
800.2.o.f 4
800.2.o.g 8
800.2.o.h 8
800.2.q $$\chi_{800}(249, \cdot)$$ None 0 2
800.2.s $$\chi_{800}(7, \cdot)$$ None 0 2
800.2.u $$\chi_{800}(161, \cdot)$$ 800.2.u.a 4 4
800.2.u.b 4
800.2.u.c 8
800.2.u.d 8
800.2.u.e 16
800.2.u.f 24
800.2.u.g 24
800.2.u.h 32
800.2.v $$\chi_{800}(43, \cdot)$$ 800.2.v.a 64 4
800.2.v.b 88
800.2.v.c 128
800.2.y $$\chi_{800}(101, \cdot)$$ 800.2.y.a 4 4
800.2.y.b 8
800.2.y.c 64
800.2.y.d 64
800.2.y.e 64
800.2.y.f 88
800.2.ba $$\chi_{800}(149, \cdot)$$ 800.2.ba.a 4 4
800.2.ba.b 4
800.2.ba.c 8
800.2.ba.d 8
800.2.ba.e 64
800.2.ba.f 64
800.2.ba.g 64
800.2.ba.h 64
800.2.bb $$\chi_{800}(107, \cdot)$$ 800.2.bb.a 64 4
800.2.bb.b 88
800.2.bb.c 128
800.2.be $$\chi_{800}(209, \cdot)$$ 800.2.be.a 112 4
800.2.bg $$\chi_{800}(129, \cdot)$$ 800.2.bg.a 8 4
800.2.bg.b 8
800.2.bg.c 8
800.2.bg.d 48
800.2.bg.e 48
800.2.bj $$\chi_{800}(81, \cdot)$$ 800.2.bj.a 112 4
800.2.bl $$\chi_{800}(23, \cdot)$$ None 0 8
800.2.bm $$\chi_{800}(41, \cdot)$$ None 0 8
800.2.bp $$\chi_{800}(47, \cdot)$$ 800.2.bp.a 8 8
800.2.bp.b 8
800.2.bp.c 208
800.2.bq $$\chi_{800}(63, \cdot)$$ 800.2.bq.a 56 8
800.2.bq.b 56
800.2.bq.c 64
800.2.bq.d 64
800.2.bt $$\chi_{800}(9, \cdot)$$ None 0 8
800.2.bu $$\chi_{800}(87, \cdot)$$ None 0 8
800.2.bx $$\chi_{800}(67, \cdot)$$ 800.2.bx.a 1888 16
800.2.by $$\chi_{800}(29, \cdot)$$ 800.2.by.a 1888 16
800.2.ca $$\chi_{800}(21, \cdot)$$ 800.2.ca.a 1888 16
800.2.cd $$\chi_{800}(3, \cdot)$$ 800.2.cd.a 1888 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(800))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(800)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 2}$$