## Defining parameters

 Level: $$N$$ = $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$384$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(48))$$.

Total New Old
Modular forms 156 59 97
Cusp forms 100 49 51
Eisenstein series 56 10 46

## Trace form

 $$49 q - q^{3} + 8 q^{4} + 12 q^{5} + 4 q^{6} + 10 q^{7} - 12 q^{8} - 11 q^{9} + O(q^{10})$$ $$49 q - q^{3} + 8 q^{4} + 12 q^{5} + 4 q^{6} + 10 q^{7} - 12 q^{8} - 11 q^{9} - 80 q^{10} + 32 q^{11} - 56 q^{12} - 34 q^{13} - 44 q^{14} - 36 q^{15} + 16 q^{16} - 12 q^{17} - 48 q^{18} - 98 q^{19} + 80 q^{20} - 26 q^{21} + 72 q^{22} - 128 q^{23} - 20 q^{24} + 33 q^{25} - 100 q^{26} + 23 q^{27} + 92 q^{29} + 156 q^{30} + 82 q^{31} + 160 q^{32} + 100 q^{33} + 232 q^{34} + 96 q^{35} + 208 q^{36} - 34 q^{37} + 168 q^{38} - 86 q^{39} + 136 q^{40} - 108 q^{41} + 220 q^{42} - 50 q^{43} + 88 q^{44} - 24 q^{45} - 64 q^{46} - 32 q^{48} + 7 q^{49} - 236 q^{50} + 128 q^{51} - 368 q^{52} - 196 q^{53} - 164 q^{54} - 192 q^{55} - 224 q^{56} - 50 q^{57} - 424 q^{58} - 128 q^{59} - 648 q^{60} - 306 q^{61} - 276 q^{62} + 90 q^{63} - 640 q^{64} - 200 q^{65} - 500 q^{66} + 318 q^{67} - 448 q^{68} + 12 q^{69} - 72 q^{70} + 512 q^{71} - 252 q^{72} + 282 q^{73} + 348 q^{74} + 459 q^{75} + 848 q^{76} + 512 q^{77} + 632 q^{78} + 370 q^{79} + 552 q^{80} - 15 q^{81} + 696 q^{82} - 160 q^{83} + 952 q^{84} + 280 q^{85} + 528 q^{86} - 96 q^{87} + 832 q^{88} + 228 q^{89} + 984 q^{90} - 28 q^{91} + 496 q^{92} - 46 q^{93} - 24 q^{94} + 8 q^{96} - 126 q^{97} - 440 q^{98} - 260 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(48))$$

We only show spaces with odd 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
48.3.b $$\chi_{48}(7, \cdot)$$ None 0 1
48.3.e $$\chi_{48}(17, \cdot)$$ 48.3.e.a 1 1
48.3.e.b 2
48.3.g $$\chi_{48}(31, \cdot)$$ 48.3.g.a 2 1
48.3.h $$\chi_{48}(41, \cdot)$$ None 0 1
48.3.i $$\chi_{48}(5, \cdot)$$ 48.3.i.a 8 2
48.3.i.b 20
48.3.l $$\chi_{48}(19, \cdot)$$ 48.3.l.a 16 2

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(48))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(48)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$