## Defining parameters

 Level: $$N$$ = $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newforms: $$4$$ Sturm bound: $$256$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(48))$$.

Total New Old
Modular forms 92 31 61
Cusp forms 37 23 14
Eisenstein series 55 8 47

## Trace form

 $$23q - q^{3} - 8q^{4} - 2q^{5} - 8q^{6} - 8q^{7} - 12q^{8} - 5q^{9} + O(q^{10})$$ $$23q - q^{3} - 8q^{4} - 2q^{5} - 8q^{6} - 8q^{7} - 12q^{8} - 5q^{9} - 8q^{10} - 12q^{11} - 10q^{13} + 12q^{14} - 10q^{15} + 16q^{16} + 2q^{17} + 8q^{18} - 16q^{19} + 16q^{20} + 4q^{21} + 16q^{22} + 8q^{23} + 28q^{24} + 9q^{25} + 20q^{26} + 11q^{27} - 10q^{29} + 20q^{30} + 16q^{31} - 8q^{33} - 8q^{34} + 24q^{35} + 16q^{36} - 34q^{37} - 8q^{38} + 18q^{39} - 24q^{40} - 6q^{41} - 44q^{42} - 40q^{44} - 14q^{45} - 48q^{46} - 64q^{48} - 45q^{49} - 36q^{50} + 34q^{51} - 32q^{52} + 14q^{53} - 48q^{54} + 32q^{55} - 8q^{57} + 16q^{58} + 28q^{59} + 8q^{60} + 54q^{61} - 12q^{62} + 8q^{63} + 64q^{64} - 12q^{65} + 52q^{66} + 16q^{67} + 32q^{68} + 28q^{69} + 72q^{70} - 8q^{71} + 36q^{72} + 30q^{73} + 52q^{74} - 19q^{75} + 64q^{76} + 16q^{77} + 48q^{78} - 16q^{79} + 8q^{80} + 7q^{81} + 24q^{82} - 36q^{83} - 8q^{84} + 12q^{85} - 16q^{86} - 54q^{87} - 32q^{88} - 6q^{89} - 24q^{90} - 64q^{91} - 16q^{92} - 16q^{93} - 40q^{94} - 56q^{95} - 56q^{96} - 34q^{97} - 40q^{98} - 64q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$

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
48.2.a $$\chi_{48}(1, \cdot)$$ 48.2.a.a 1 1
48.2.c $$\chi_{48}(47, \cdot)$$ 48.2.c.a 2 1
48.2.d $$\chi_{48}(25, \cdot)$$ None 0 1
48.2.f $$\chi_{48}(23, \cdot)$$ None 0 1
48.2.j $$\chi_{48}(13, \cdot)$$ 48.2.j.a 8 2
48.2.k $$\chi_{48}(11, \cdot)$$ 48.2.k.a 12 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(48))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(48)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$