## Defining parameters

 Level: $$N$$ = $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ 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

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

## 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

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