## Defining parameters

 Level: $$N$$ = $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$3$$ Newform subspaces: $$3$$ Sturm bound: $$96$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(30))$$.

Total New Old
Modular forms 40 7 33
Cusp forms 9 7 2
Eisenstein series 31 0 31

## Trace form

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

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$

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
30.2.a $$\chi_{30}(1, \cdot)$$ 30.2.a.a 1 1
30.2.c $$\chi_{30}(19, \cdot)$$ 30.2.c.a 2 1
30.2.e $$\chi_{30}(17, \cdot)$$ 30.2.e.a 4 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(30))$$ into lower level spaces

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