## Defining parameters

 Level: $$N$$ = $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$27$$ Sturm bound: $$9600$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(300))$$.

Total New Old
Modular forms 2680 1010 1670
Cusp forms 2121 926 1195
Eisenstein series 559 84 475

## Trace form

 $$926 q - 4 q^{3} - 4 q^{4} - 2 q^{5} - 2 q^{6} + 8 q^{7} + 24 q^{8} + O(q^{10})$$ $$926 q - 4 q^{3} - 4 q^{4} - 2 q^{5} - 2 q^{6} + 8 q^{7} + 24 q^{8} + 16 q^{11} - 2 q^{12} + 8 q^{13} + 14 q^{15} - 36 q^{16} + 60 q^{17} - 42 q^{18} + 40 q^{19} - 20 q^{20} + 8 q^{21} - 36 q^{22} + 40 q^{23} - 68 q^{24} + 34 q^{25} - 32 q^{26} - 28 q^{27} - 68 q^{28} + 16 q^{29} - 46 q^{30} - 8 q^{31} - 40 q^{32} - 24 q^{33} - 68 q^{34} + 4 q^{35} - 50 q^{36} - 50 q^{37} - 92 q^{38} - 40 q^{39} - 136 q^{40} - 32 q^{41} - 126 q^{42} - 56 q^{43} - 140 q^{44} - 98 q^{45} - 108 q^{46} - 80 q^{47} - 100 q^{48} - 134 q^{49} - 164 q^{50} - 36 q^{51} - 120 q^{52} - 98 q^{53} - 28 q^{54} - 72 q^{55} + 8 q^{56} - 140 q^{57} - 48 q^{58} - 64 q^{59} - 46 q^{60} - 40 q^{61} + 12 q^{62} - 106 q^{63} + 32 q^{64} - 30 q^{65} + 62 q^{66} + 8 q^{67} + 32 q^{68} - 154 q^{69} - 12 q^{70} - 34 q^{72} - 136 q^{73} - 82 q^{75} - 56 q^{76} - 96 q^{77} - 72 q^{78} - 48 q^{79} - 4 q^{80} - 128 q^{81} - 52 q^{82} - 40 q^{83} - 42 q^{84} - 246 q^{85} - 8 q^{86} - 130 q^{87} + 12 q^{88} - 310 q^{89} - 10 q^{90} - 128 q^{91} + 88 q^{92} - 198 q^{93} + 92 q^{94} - 80 q^{95} - 46 q^{96} - 416 q^{97} + 232 q^{98} - 16 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(300))$$

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
300.2.a $$\chi_{300}(1, \cdot)$$ 300.2.a.a 1 1
300.2.a.b 1
300.2.a.c 1
300.2.a.d 1
300.2.d $$\chi_{300}(49, \cdot)$$ 300.2.d.a 2 1
300.2.e $$\chi_{300}(251, \cdot)$$ 300.2.e.a 4 1
300.2.e.b 4
300.2.e.c 8
300.2.e.d 8
300.2.e.e 8
300.2.h $$\chi_{300}(299, \cdot)$$ 300.2.h.a 8 1
300.2.h.b 8
300.2.h.c 16
300.2.i $$\chi_{300}(257, \cdot)$$ 300.2.i.a 4 2
300.2.i.b 4
300.2.i.c 4
300.2.j $$\chi_{300}(7, \cdot)$$ 300.2.j.a 8 2
300.2.j.b 8
300.2.j.c 8
300.2.j.d 12
300.2.m $$\chi_{300}(61, \cdot)$$ 300.2.m.a 8 4
300.2.m.b 8
300.2.n $$\chi_{300}(11, \cdot)$$ 300.2.n.a 224 4
300.2.o $$\chi_{300}(109, \cdot)$$ 300.2.o.a 24 4
300.2.r $$\chi_{300}(59, \cdot)$$ 300.2.r.a 224 4
300.2.w $$\chi_{300}(67, \cdot)$$ 300.2.w.a 240 8
300.2.x $$\chi_{300}(17, \cdot)$$ 300.2.x.a 80 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(300))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(300)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 2}$$