## 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

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

## 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

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