## Defining parameters

 Level: $$N$$ = $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newforms: $$10$$ Sturm bound: $$1200$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(100))$$.

Total New Old
Modular forms 370 181 189
Cusp forms 231 141 90
Eisenstein series 139 40 99

## Trace form

 $$141q - 6q^{2} + 4q^{3} - 10q^{4} - 15q^{5} - 18q^{6} - 4q^{7} - 18q^{8} - 22q^{9} + O(q^{10})$$ $$141q - 6q^{2} + 4q^{3} - 10q^{4} - 15q^{5} - 18q^{6} - 4q^{7} - 18q^{8} - 22q^{9} - 16q^{10} - 10q^{12} - 20q^{13} - 10q^{14} - 2q^{15} - 2q^{16} - 30q^{17} + 2q^{18} - 12q^{19} - 6q^{20} - 48q^{21} - 10q^{22} - 32q^{23} - 57q^{25} - 36q^{26} - 38q^{27} - 10q^{28} - 52q^{29} - 10q^{30} - 12q^{31} - 26q^{32} - 40q^{33} - 10q^{34} - 8q^{35} - 10q^{36} + 9q^{37} + 20q^{38} + 48q^{39} + 44q^{40} + 4q^{41} + 90q^{42} + 60q^{43} + 60q^{44} + 55q^{45} + 42q^{46} + 52q^{47} + 120q^{48} + 51q^{49} + 94q^{50} + 36q^{51} + 88q^{52} + q^{53} + 120q^{54} + 40q^{55} + 42q^{56} + 44q^{57} + 44q^{58} + 6q^{59} + 90q^{60} - 88q^{61} + 40q^{62} - 14q^{63} + 20q^{64} - 65q^{65} + 30q^{66} - 44q^{67} + 14q^{68} - 66q^{69} - 10q^{70} - 16q^{71} - 16q^{72} - 20q^{73} - 78q^{75} - 60q^{76} - 100q^{77} + 20q^{78} - 56q^{79} - 26q^{80} + 54q^{81} - 82q^{82} - 62q^{83} - 90q^{84} + 27q^{85} - 78q^{86} + 14q^{87} - 130q^{88} + 77q^{89} - 166q^{90} + 12q^{91} - 110q^{92} + 74q^{93} - 170q^{94} + 36q^{95} - 118q^{96} + 144q^{97} - 158q^{98} + 60q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$

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
100.2.a $$\chi_{100}(1, \cdot)$$ 100.2.a.a 1 1
100.2.c $$\chi_{100}(49, \cdot)$$ 100.2.c.a 2 1
100.2.e $$\chi_{100}(7, \cdot)$$ 100.2.e.a 2 2
100.2.e.b 2
100.2.e.c 2
100.2.e.d 8
100.2.g $$\chi_{100}(21, \cdot)$$ 100.2.g.a 12 4
100.2.i $$\chi_{100}(9, \cdot)$$ 100.2.i.a 8 4
100.2.l $$\chi_{100}(3, \cdot)$$ 100.2.l.a 8 8
100.2.l.b 96

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(100))$$ into lower level spaces

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