## Defining parameters

 Level: $$N$$ = $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$14$$ Newform subspaces: $$40$$ Sturm bound: $$6144$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(240))$$.

Total New Old
Modular forms 1760 622 1138
Cusp forms 1313 566 747
Eisenstein series 447 56 391

## Trace form

 $$566q - 4q^{3} + 2q^{5} + 4q^{7} + 24q^{8} + 10q^{9} + O(q^{10})$$ $$566q - 4q^{3} + 2q^{5} + 4q^{7} + 24q^{8} + 10q^{9} - 8q^{10} + 24q^{11} - 8q^{12} + 8q^{13} - 24q^{14} + 16q^{15} - 64q^{16} + 12q^{17} - 24q^{18} + 40q^{19} - 16q^{20} - 12q^{21} - 48q^{22} + 8q^{23} - 64q^{24} - 2q^{25} - 40q^{26} - 16q^{27} - 80q^{28} + 12q^{29} - 60q^{30} - 72q^{31} - 80q^{32} - 28q^{33} - 128q^{34} - 48q^{35} - 64q^{36} - 32q^{37} - 96q^{38} - 68q^{39} - 152q^{40} - 52q^{41} - 44q^{43} - 32q^{44} - 22q^{45} - 64q^{46} - 24q^{47} + 56q^{48} - 6q^{49} - 64q^{50} - 112q^{51} - 16q^{52} - 44q^{53} + 88q^{54} - 64q^{55} - 12q^{57} - 48q^{58} - 56q^{59} + 16q^{60} - 148q^{61} + 24q^{62} - 128q^{63} - 96q^{64} + 4q^{65} - 56q^{66} - 164q^{67} + 48q^{68} - 112q^{69} + 56q^{70} - 144q^{71} + 40q^{72} - 56q^{73} + 72q^{74} - 144q^{75} - 32q^{76} - 80q^{77} + 64q^{78} - 168q^{79} + 152q^{80} - 66q^{81} + 64q^{82} - 104q^{83} + 160q^{84} - 40q^{85} + 160q^{86} - 8q^{87} + 224q^{88} + 52q^{89} + 136q^{90} + 40q^{91} + 176q^{92} + 24q^{93} + 176q^{94} - 16q^{95} + 160q^{96} + 88q^{97} + 128q^{98} + 48q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(240))$$

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
240.2.a $$\chi_{240}(1, \cdot)$$ 240.2.a.a 1 1
240.2.a.b 1
240.2.a.c 1
240.2.a.d 1
240.2.b $$\chi_{240}(71, \cdot)$$ None 0 1
240.2.d $$\chi_{240}(169, \cdot)$$ None 0 1
240.2.f $$\chi_{240}(49, \cdot)$$ 240.2.f.a 2 1
240.2.f.b 2
240.2.f.c 2
240.2.h $$\chi_{240}(191, \cdot)$$ 240.2.h.a 4 1
240.2.h.b 4
240.2.k $$\chi_{240}(121, \cdot)$$ None 0 1
240.2.m $$\chi_{240}(119, \cdot)$$ None 0 1
240.2.o $$\chi_{240}(239, \cdot)$$ 240.2.o.a 4 1
240.2.o.b 8
240.2.s $$\chi_{240}(61, \cdot)$$ 240.2.s.a 4 2
240.2.s.b 8
240.2.s.c 20
240.2.t $$\chi_{240}(59, \cdot)$$ 240.2.t.a 8 2
240.2.t.b 80
240.2.v $$\chi_{240}(17, \cdot)$$ 240.2.v.a 4 2
240.2.v.b 4
240.2.v.c 4
240.2.v.d 4
240.2.v.e 4
240.2.w $$\chi_{240}(127, \cdot)$$ 240.2.w.a 4 2
240.2.w.b 8
240.2.y $$\chi_{240}(163, \cdot)$$ 240.2.y.a 2 2
240.2.y.b 2
240.2.y.c 2
240.2.y.d 6
240.2.y.e 16
240.2.y.f 20
240.2.bb $$\chi_{240}(173, \cdot)$$ 240.2.bb.a 88 2
240.2.bc $$\chi_{240}(43, \cdot)$$ 240.2.bc.a 2 2
240.2.bc.b 2
240.2.bc.c 2
240.2.bc.d 6
240.2.bc.e 16
240.2.bc.f 20
240.2.bf $$\chi_{240}(53, \cdot)$$ 240.2.bf.a 88 2
240.2.bh $$\chi_{240}(7, \cdot)$$ None 0 2
240.2.bi $$\chi_{240}(137, \cdot)$$ None 0 2
240.2.bk $$\chi_{240}(11, \cdot)$$ 240.2.bk.a 4 2
240.2.bk.b 60
240.2.bl $$\chi_{240}(109, \cdot)$$ 240.2.bl.a 48 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(240))$$ into lower level spaces

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