## 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

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

## 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

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