## Defining parameters

 Level: $$N$$ = $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$12$$ Sturm bound: $$2400$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(150))$$.

Total New Old
Modular forms 712 137 575
Cusp forms 489 137 352
Eisenstein series 223 0 223

## Trace form

 $$137q + 3q^{2} + 7q^{3} + 3q^{4} + 10q^{5} + 7q^{6} + 24q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$137q + 3q^{2} + 7q^{3} + 3q^{4} + 10q^{5} + 7q^{6} + 24q^{7} + 3q^{8} + 3q^{9} + 2q^{10} + 4q^{11} - 9q^{12} + 10q^{13} - 8q^{14} - 4q^{15} + 3q^{16} - 34q^{17} - 23q^{18} - 52q^{19} - 8q^{20} - 32q^{21} - 60q^{22} - 56q^{23} - 13q^{24} - 86q^{25} + 2q^{26} + 7q^{27} - 16q^{28} - 38q^{29} - 20q^{30} - 16q^{31} - 7q^{32} - 20q^{33} - 28q^{34} - 8q^{35} + 3q^{36} - 24q^{37} + 12q^{38} - 54q^{39} - 6q^{40} + 6q^{41} - 8q^{42} - 76q^{43} + 20q^{44} - 70q^{45} - 24q^{46} - 32q^{47} - 9q^{48} - 63q^{49} + 18q^{50} - 30q^{51} + 10q^{52} - 24q^{53} + 7q^{54} - 24q^{55} + 8q^{56} - 20q^{57} + 58q^{58} + 20q^{59} + 32q^{60} + 82q^{61} + 48q^{62} + 84q^{63} + 3q^{64} + 74q^{65} + 48q^{66} + 108q^{67} + 6q^{68} + 180q^{69} + 48q^{70} + 24q^{71} + 19q^{72} + 110q^{73} + 34q^{74} + 164q^{75} - 4q^{76} + 96q^{77} + 118q^{78} + 64q^{79} + 10q^{80} + 19q^{81} + 62q^{82} + 20q^{83} + 68q^{84} + 34q^{85} + 20q^{86} + 62q^{87} + 20q^{88} + 4q^{89} + 54q^{90} + 24q^{92} - 20q^{93} + 80q^{94} + 16q^{95} - 9q^{96} + 30q^{97} + 75q^{98} + 20q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$

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
150.2.a $$\chi_{150}(1, \cdot)$$ 150.2.a.a 1 1
150.2.a.b 1
150.2.a.c 1
150.2.c $$\chi_{150}(49, \cdot)$$ 150.2.c.a 2 1
150.2.e $$\chi_{150}(107, \cdot)$$ 150.2.e.a 4 2
150.2.e.b 8
150.2.g $$\chi_{150}(31, \cdot)$$ 150.2.g.a 4 4
150.2.g.b 4
150.2.g.c 8
150.2.h $$\chi_{150}(19, \cdot)$$ 150.2.h.a 8 4
150.2.h.b 16
150.2.l $$\chi_{150}(17, \cdot)$$ 150.2.l.a 80 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(150))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(150)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 2}$$