## Defining parameters

 Level: $$N$$ = $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$4$$ 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

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

## 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

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