## Defining parameters

 Level: $$N$$ = $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$14$$ Newform subspaces: $$43$$ Sturm bound: $$12288$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(240))$$.

Total New Old
Modular forms 4832 1810 3022
Cusp forms 4384 1754 2630
Eisenstein series 448 56 392

## Trace form

 $$1754 q + 4 q^{3} + 32 q^{4} - 2 q^{5} - 72 q^{6} - 60 q^{7} - 168 q^{8} - 114 q^{9} + O(q^{10})$$ $$1754 q + 4 q^{3} + 32 q^{4} - 2 q^{5} - 72 q^{6} - 60 q^{7} - 168 q^{8} - 114 q^{9} - 144 q^{10} + 120 q^{11} + 200 q^{12} + 192 q^{13} + 696 q^{14} - 144 q^{15} + 576 q^{16} - 156 q^{17} - 40 q^{18} - 536 q^{19} - 80 q^{20} + 228 q^{21} - 1344 q^{22} + 328 q^{23} - 224 q^{24} + 18 q^{25} + 40 q^{26} + 760 q^{27} + 240 q^{28} - 684 q^{29} - 900 q^{30} + 888 q^{31} + 560 q^{32} - 1180 q^{33} + 1504 q^{34} + 336 q^{35} + 1408 q^{36} + 392 q^{37} + 1632 q^{38} - 1316 q^{39} + 3848 q^{40} + 1540 q^{41} - 80 q^{42} + 52 q^{43} + 2528 q^{44} + 1270 q^{45} + 1120 q^{46} + 744 q^{47} - 3464 q^{48} - 450 q^{49} - 5888 q^{50} + 96 q^{51} - 10160 q^{52} - 4756 q^{53} - 3408 q^{54} - 1760 q^{55} - 2688 q^{56} - 2812 q^{57} - 3040 q^{58} + 1256 q^{59} + 928 q^{60} + 1332 q^{61} + 1992 q^{62} - 1312 q^{63} + 11936 q^{64} - 1396 q^{65} + 10648 q^{66} + 92 q^{67} + 9072 q^{68} + 6896 q^{69} + 8520 q^{70} + 3696 q^{71} + 5864 q^{72} + 9344 q^{73} - 3144 q^{74} + 3976 q^{75} - 3712 q^{76} - 1840 q^{77} - 5168 q^{78} + 3736 q^{79} - 13352 q^{80} + 466 q^{81} - 13280 q^{82} + 5816 q^{83} - 8384 q^{84} + 1600 q^{85} - 1312 q^{86} - 344 q^{87} - 11104 q^{88} + 572 q^{89} + 392 q^{90} - 2776 q^{91} + 496 q^{92} + 1704 q^{93} - 7120 q^{94} - 12176 q^{95} - 9440 q^{96} - 6480 q^{97} - 896 q^{98} - 7152 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(240))$$

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
240.4.a $$\chi_{240}(1, \cdot)$$ 240.4.a.a 1 1
240.4.a.b 1
240.4.a.c 1
240.4.a.d 1
240.4.a.e 1
240.4.a.f 1
240.4.a.g 1
240.4.a.h 1
240.4.a.i 1
240.4.a.j 1
240.4.a.k 1
240.4.a.l 1
240.4.b $$\chi_{240}(71, \cdot)$$ None 0 1
240.4.d $$\chi_{240}(169, \cdot)$$ None 0 1
240.4.f $$\chi_{240}(49, \cdot)$$ 240.4.f.a 2 1
240.4.f.b 2
240.4.f.c 2
240.4.f.d 2
240.4.f.e 2
240.4.f.f 4
240.4.f.g 4
240.4.h $$\chi_{240}(191, \cdot)$$ 240.4.h.a 8 1
240.4.h.b 16
240.4.k $$\chi_{240}(121, \cdot)$$ None 0 1
240.4.m $$\chi_{240}(119, \cdot)$$ None 0 1
240.4.o $$\chi_{240}(239, \cdot)$$ 240.4.o.a 4 1
240.4.o.b 8
240.4.o.c 24
240.4.s $$\chi_{240}(61, \cdot)$$ 240.4.s.a 44 2
240.4.s.b 52
240.4.t $$\chi_{240}(59, \cdot)$$ 240.4.t.a 8 2
240.4.t.b 272
240.4.v $$\chi_{240}(17, \cdot)$$ 240.4.v.a 4 2
240.4.v.b 8
240.4.v.c 8
240.4.v.d 12
240.4.v.e 36
240.4.w $$\chi_{240}(127, \cdot)$$ 240.4.w.a 12 2
240.4.w.b 24
240.4.y $$\chi_{240}(163, \cdot)$$ 240.4.y.a 72 2
240.4.y.b 72
240.4.bb $$\chi_{240}(173, \cdot)$$ 240.4.bb.a 280 2
240.4.bc $$\chi_{240}(43, \cdot)$$ 240.4.bc.a 72 2
240.4.bc.b 72
240.4.bf $$\chi_{240}(53, \cdot)$$ 240.4.bf.a 280 2
240.4.bh $$\chi_{240}(7, \cdot)$$ None 0 2
240.4.bi $$\chi_{240}(137, \cdot)$$ None 0 2
240.4.bk $$\chi_{240}(11, \cdot)$$ 240.4.bk.a 192 2
240.4.bl $$\chi_{240}(109, \cdot)$$ 240.4.bl.a 144 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(240))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(240)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$