## Defining parameters

 Level: $$N$$ = $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$10$$ Newform subspaces: $$35$$ Sturm bound: $$4800$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(200))$$.

Total New Old
Modular forms 1368 663 705
Cusp forms 1033 581 452
Eisenstein series 335 82 253

## Trace form

 $$581q - 12q^{2} - 12q^{3} - 12q^{4} + q^{5} - 20q^{6} - 4q^{7} - 12q^{8} - 14q^{9} + O(q^{10})$$ $$581q - 12q^{2} - 12q^{3} - 12q^{4} + q^{5} - 20q^{6} - 4q^{7} - 12q^{8} - 14q^{9} - 16q^{10} - 12q^{11} - 36q^{12} + 4q^{13} - 36q^{14} - 24q^{15} - 52q^{16} - 28q^{17} - 68q^{18} - 44q^{19} - 36q^{20} - 16q^{21} - 52q^{22} - 20q^{23} - 68q^{24} - 27q^{25} - 72q^{26} - 36q^{27} - 4q^{28} + 12q^{29} - 20q^{30} - 36q^{31} + 28q^{32} - 24q^{33} + 52q^{34} - 28q^{35} + 44q^{36} - 17q^{37} + 28q^{38} - 76q^{39} + 24q^{40} - 56q^{41} + 28q^{42} - 100q^{43} + 12q^{44} - 79q^{45} + 28q^{46} - 108q^{47} - 4q^{48} - 99q^{49} - 36q^{50} - 132q^{51} - 60q^{52} - 57q^{53} - 68q^{54} - 88q^{55} - 132q^{56} - 152q^{57} - 108q^{58} - 100q^{59} - 60q^{60} + 24q^{61} - 132q^{62} - 44q^{63} - 84q^{64} - 59q^{65} - 68q^{66} + 20q^{67} - 60q^{68} - 16q^{69} - 20q^{70} + 12q^{71} + 64q^{72} - 92q^{73} + 20q^{74} + 56q^{75} + 8q^{76} + 32q^{77} + 108q^{78} + 140q^{79} + 4q^{80} - 78q^{81} + 132q^{82} + 160q^{83} + 204q^{84} + 73q^{85} + 68q^{86} + 196q^{87} + 220q^{88} - 53q^{89} + 284q^{90} + 116q^{91} + 212q^{92} + 120q^{93} + 236q^{94} + 112q^{95} + 68q^{96} + 48q^{97} + 236q^{98} + 188q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$

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
200.2.a $$\chi_{200}(1, \cdot)$$ 200.2.a.a 1 1
200.2.a.b 1
200.2.a.c 1
200.2.a.d 1
200.2.a.e 1
200.2.c $$\chi_{200}(49, \cdot)$$ 200.2.c.a 2 1
200.2.c.b 2
200.2.d $$\chi_{200}(101, \cdot)$$ 200.2.d.a 2 1
200.2.d.b 2
200.2.d.c 2
200.2.d.d 2
200.2.d.e 4
200.2.d.f 4
200.2.f $$\chi_{200}(149, \cdot)$$ 200.2.f.a 2 1
200.2.f.b 2
200.2.f.c 4
200.2.f.d 4
200.2.f.e 4
200.2.j $$\chi_{200}(7, \cdot)$$ None 0 2
200.2.k $$\chi_{200}(43, \cdot)$$ 200.2.k.a 2 2
200.2.k.b 2
200.2.k.c 2
200.2.k.d 2
200.2.k.e 4
200.2.k.f 4
200.2.k.g 8
200.2.k.h 8
200.2.m $$\chi_{200}(41, \cdot)$$ 200.2.m.a 4 4
200.2.m.b 8
200.2.m.c 16
200.2.o $$\chi_{200}(29, \cdot)$$ 200.2.o.a 112 4
200.2.q $$\chi_{200}(9, \cdot)$$ 200.2.q.a 32 4
200.2.t $$\chi_{200}(21, \cdot)$$ 200.2.t.a 112 4
200.2.v $$\chi_{200}(3, \cdot)$$ 200.2.v.a 8 8
200.2.v.b 8
200.2.v.c 208
200.2.w $$\chi_{200}(23, \cdot)$$ None 0 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(200))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(200)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$