## Defining parameters

 Level: $$N$$ = $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$19$$ Sturm bound: $$1440$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(99))$$.

Total New Old
Modular forms 440 320 120
Cusp forms 281 240 41
Eisenstein series 159 80 79

## Trace form

 $$240 q - 15 q^{2} - 20 q^{3} - 15 q^{4} - 15 q^{5} - 20 q^{6} - 20 q^{7} - 25 q^{8} - 20 q^{9} + O(q^{10})$$ $$240 q - 15 q^{2} - 20 q^{3} - 15 q^{4} - 15 q^{5} - 20 q^{6} - 20 q^{7} - 25 q^{8} - 20 q^{9} - 60 q^{10} - 20 q^{11} - 40 q^{12} - 20 q^{13} - 30 q^{14} - 20 q^{15} - 45 q^{16} - 30 q^{17} - 20 q^{18} - 60 q^{19} - 10 q^{20} - 20 q^{21} - 45 q^{22} - 25 q^{23} + 10 q^{24} - 5 q^{25} + 20 q^{26} + 10 q^{27} + 20 q^{28} + 20 q^{29} + 40 q^{30} + 5 q^{31} + 95 q^{32} + 30 q^{33} + 10 q^{34} + 40 q^{35} + 50 q^{36} - 35 q^{37} + 50 q^{38} + 10 q^{39} + 20 q^{40} + 40 q^{42} - 20 q^{43} - 25 q^{44} - 30 q^{45} - 110 q^{46} - 50 q^{47} + 20 q^{48} - 70 q^{49} + 5 q^{50} - 10 q^{51} - 20 q^{52} + 10 q^{53} + 60 q^{54} - 25 q^{55} + 60 q^{56} + 60 q^{57} + 60 q^{58} + 75 q^{59} + 120 q^{60} + 20 q^{61} + 180 q^{62} + 60 q^{63} + 105 q^{64} + 90 q^{65} + 110 q^{66} + 65 q^{67} + 160 q^{68} + 60 q^{69} + 90 q^{70} + 75 q^{71} + 130 q^{72} - 30 q^{73} + 150 q^{74} + 50 q^{75} + 10 q^{76} + 60 q^{77} + 60 q^{78} - 20 q^{79} - 120 q^{80} - 20 q^{81} - 10 q^{82} - 110 q^{83} - 110 q^{84} - 80 q^{85} - 160 q^{86} - 120 q^{87} - 95 q^{88} - 145 q^{89} - 220 q^{90} - 40 q^{91} - 220 q^{92} - 120 q^{93} - 90 q^{94} - 210 q^{95} - 310 q^{96} - 75 q^{97} - 275 q^{98} - 180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(99))$$

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
99.2.a $$\chi_{99}(1, \cdot)$$ 99.2.a.a 1 1
99.2.a.b 1
99.2.a.c 1
99.2.a.d 1
99.2.d $$\chi_{99}(98, \cdot)$$ 99.2.d.a 4 1
99.2.e $$\chi_{99}(34, \cdot)$$ 99.2.e.a 2 2
99.2.e.b 2
99.2.e.c 2
99.2.e.d 6
99.2.e.e 8
99.2.f $$\chi_{99}(37, \cdot)$$ 99.2.f.a 4 4
99.2.f.b 4
99.2.f.c 8
99.2.g $$\chi_{99}(32, \cdot)$$ 99.2.g.a 4 2
99.2.g.b 16
99.2.j $$\chi_{99}(8, \cdot)$$ 99.2.j.a 16 4
99.2.m $$\chi_{99}(4, \cdot)$$ 99.2.m.a 8 8
99.2.m.b 72
99.2.p $$\chi_{99}(2, \cdot)$$ 99.2.p.a 80 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(99))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(99)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$