## 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

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

## 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

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