## Defining parameters

 Level: $$N$$ = $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ Sturm bound: $$512$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(48))$$.

Total New Old
Modular forms 220 85 135
Cusp forms 164 77 87
Eisenstein series 56 8 48

## Trace form

 $$77q - 5q^{3} - 24q^{4} + 2q^{5} + 28q^{6} + 24q^{7} + 84q^{8} + 57q^{9} + O(q^{10})$$ $$77q - 5q^{3} - 24q^{4} + 2q^{5} + 28q^{6} + 24q^{7} + 84q^{8} + 57q^{9} + 128q^{10} - 60q^{11} - 104q^{12} - 14q^{13} - 348q^{14} + 150q^{15} - 304q^{16} - 26q^{17} + 16q^{18} + 32q^{19} + 80q^{20} - 260q^{21} + 664q^{22} - 248q^{23} + 108q^{24} - 221q^{25} - 20q^{26} - 161q^{27} - 640q^{28} + 58q^{29} - 740q^{30} - 720q^{31} - 960q^{32} + 560q^{33} - 184q^{34} + 120q^{35} - 496q^{36} + 1114q^{37} + 1256q^{38} + 962q^{39} + 2376q^{40} + 414q^{41} + 1356q^{42} + 400q^{43} - 200q^{44} - 1386q^{45} - 768q^{46} - 576q^{47} + 672q^{48} - 1423q^{49} + 708q^{50} - 582q^{51} + 1328q^{52} + 1234q^{53} + 1700q^{54} + 1248q^{55} + 1344q^{56} + 2280q^{57} + 1512q^{58} - 628q^{59} - 424q^{60} - 686q^{61} - 996q^{62} - 216q^{63} - 2112q^{64} + 348q^{65} - 2036q^{66} - 4672q^{67} - 1568q^{68} - 4196q^{69} - 6952q^{70} - 1288q^{71} - 4316q^{72} - 4822q^{73} - 2740q^{74} + 1481q^{75} - 2448q^{76} + 2672q^{77} - 3240q^{78} + 6032q^{79} + 712q^{80} + 365q^{81} + 728q^{82} + 5868q^{83} - 296q^{84} + 5172q^{85} - 1712q^{86} + 2490q^{87} + 4928q^{88} - 66q^{89} + 3320q^{90} - 4160q^{91} + 5296q^{92} - 3032q^{93} + 10888q^{94} - 10936q^{95} + 10888q^{96} - 662q^{97} + 6760q^{98} + 656q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$

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
48.4.a $$\chi_{48}(1, \cdot)$$ 48.4.a.a 1 1
48.4.a.b 1
48.4.a.c 1
48.4.c $$\chi_{48}(47, \cdot)$$ 48.4.c.a 2 1
48.4.c.b 4
48.4.d $$\chi_{48}(25, \cdot)$$ None 0 1
48.4.f $$\chi_{48}(23, \cdot)$$ None 0 1
48.4.j $$\chi_{48}(13, \cdot)$$ 48.4.j.a 24 2
48.4.k $$\chi_{48}(11, \cdot)$$ 48.4.k.a 44 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(48))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(48)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$