## Defining parameters

 Level: $$N$$ = $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$3$$ Sturm bound: $$144$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(30))$$.

Total New Old
Modular forms 64 12 52
Cusp forms 32 12 20
Eisenstein series 32 0 32

## Trace form

 $$12q + 4q^{2} + 4q^{3} - 8q^{6} - 8q^{7} - 8q^{8} - 40q^{9} + O(q^{10})$$ $$12q + 4q^{2} + 4q^{3} - 8q^{6} - 8q^{7} - 8q^{8} - 40q^{9} - 12q^{10} - 16q^{11} - 8q^{12} - 28q^{13} + 4q^{15} + 16q^{16} + 44q^{17} + 44q^{18} + 80q^{19} + 8q^{20} + 64q^{21} + 32q^{22} - 16q^{23} + 36q^{25} + 24q^{26} + 28q^{27} + 16q^{28} - 176q^{31} - 16q^{32} - 24q^{33} - 160q^{34} - 112q^{35} - 24q^{36} - 196q^{37} - 64q^{38} - 40q^{39} - 24q^{40} + 32q^{41} - 80q^{42} + 104q^{43} - 60q^{45} + 80q^{46} + 96q^{47} + 16q^{48} + 240q^{49} + 92q^{50} + 152q^{51} + 104q^{52} + 100q^{53} + 240q^{54} + 464q^{55} + 64q^{56} + 200q^{57} + 64q^{58} - 72q^{60} - 224q^{61} - 80q^{62} - 304q^{63} - 204q^{65} - 320q^{66} - 344q^{67} - 88q^{68} - 200q^{69} - 48q^{70} + 64q^{71} - 40q^{72} + 12q^{73} - 244q^{75} - 96q^{76} - 128q^{77} + 136q^{78} - 400q^{79} + 180q^{81} + 224q^{82} + 192q^{83} + 240q^{84} - 44q^{85} + 96q^{86} + 192q^{87} - 64q^{88} + 36q^{90} + 208q^{91} - 32q^{92} - 64q^{93} - 320q^{94} + 64q^{95} - 32q^{96} + 428q^{97} - 124q^{98} + 80q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$

We only show spaces with odd 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
30.3.b $$\chi_{30}(29, \cdot)$$ 30.3.b.a 4 1
30.3.d $$\chi_{30}(11, \cdot)$$ 30.3.d.a 4 1
30.3.f $$\chi_{30}(7, \cdot)$$ 30.3.f.a 4 2

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(30))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(30)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$