## Defining parameters

 Level: $$N$$ = $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$25$$ Sturm bound: $$4800$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(150))$$.

Total New Old
Modular forms 1912 415 1497
Cusp forms 1688 415 1273
Eisenstein series 224 0 224

## Trace form

 $$415q + 6q^{2} - 19q^{3} - 12q^{4} - 10q^{5} - 10q^{6} - 32q^{7} + 24q^{8} - 27q^{9} + O(q^{10})$$ $$415q + 6q^{2} - 19q^{3} - 12q^{4} - 10q^{5} - 10q^{6} - 32q^{7} + 24q^{8} - 27q^{9} + 52q^{10} - 100q^{11} + 52q^{12} + 190q^{13} + 32q^{14} + 256q^{15} + 208q^{16} - 758q^{17} - 94q^{18} - 572q^{19} + 32q^{20} - 784q^{21} + 840q^{22} + 1112q^{23} + 312q^{24} + 2774q^{25} + 532q^{26} + 1349q^{27} + 352q^{28} + 662q^{29} - 360q^{30} - 1080q^{31} - 224q^{32} - 2740q^{33} - 3608q^{34} - 3568q^{35} - 1132q^{36} - 2968q^{37} - 744q^{38} - 2554q^{39} + 144q^{40} - 294q^{41} + 2464q^{42} + 6012q^{43} + 1840q^{44} + 3630q^{45} + 3824q^{46} + 4016q^{47} + 208q^{48} + 2607q^{49} - 252q^{50} + 1342q^{51} - 1160q^{52} + 2448q^{53} - 378q^{54} - 344q^{55} + 256q^{56} - 2700q^{57} - 5196q^{58} - 4580q^{59} + 1632q^{60} - 2826q^{61} - 2544q^{62} - 2128q^{63} - 192q^{64} - 3146q^{65} - 688q^{66} - 1964q^{67} + 2088q^{68} - 3880q^{69} + 2688q^{70} + 1080q^{71} + 88q^{72} - 1550q^{73} + 1364q^{74} - 11256q^{75} + 1168q^{76} + 4032q^{77} + 844q^{78} + 5704q^{79} - 160q^{80} + 5181q^{81} - 2436q^{82} - 3140q^{83} - 192q^{84} - 4186q^{85} - 3224q^{86} - 2346q^{87} - 2400q^{88} - 5356q^{89} + 4684q^{90} - 1552q^{91} - 2592q^{92} + 5440q^{93} - 5440q^{94} + 7376q^{95} + 352q^{96} + 18130q^{97} + 3510q^{98} + 4140q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(150))$$

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
150.4.a $$\chi_{150}(1, \cdot)$$ 150.4.a.a 1 1
150.4.a.b 1
150.4.a.c 1
150.4.a.d 1
150.4.a.e 1
150.4.a.f 1
150.4.a.g 1
150.4.a.h 1
150.4.a.i 1
150.4.c $$\chi_{150}(49, \cdot)$$ 150.4.c.a 2 1
150.4.c.b 2
150.4.c.c 2
150.4.c.d 2
150.4.c.e 2
150.4.e $$\chi_{150}(107, \cdot)$$ 150.4.e.a 4 2
150.4.e.b 4
150.4.e.c 12
150.4.e.d 16
150.4.g $$\chi_{150}(31, \cdot)$$ 150.4.g.a 12 4
150.4.g.b 16
150.4.g.c 16
150.4.g.d 20
150.4.h $$\chi_{150}(19, \cdot)$$ 150.4.h.a 24 4
150.4.h.b 32
150.4.l $$\chi_{150}(17, \cdot)$$ 150.4.l.a 240 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(150))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(150)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 2}$$