## 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

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

## 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

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