# Properties

 Label 240.3 Level 240 Weight 3 Dimension 1162 Nonzero newspaces 14 Newform subspaces 31 Sturm bound 9216 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$14$$ Newform subspaces: $$31$$ Sturm bound: $$9216$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(240))$$.

Total New Old
Modular forms 3296 1214 2082
Cusp forms 2848 1162 1686
Eisenstein series 448 52 396

## Trace form

 $$1162 q - 4 q^{3} - 32 q^{4} - 12 q^{5} - 24 q^{6} - 32 q^{7} + 24 q^{8} + 18 q^{9} + O(q^{10})$$ $$1162 q - 4 q^{3} - 32 q^{4} - 12 q^{5} - 24 q^{6} - 32 q^{7} + 24 q^{8} + 18 q^{9} + 64 q^{10} - 64 q^{11} + 104 q^{12} + 88 q^{14} - 18 q^{15} - 64 q^{16} - 8 q^{17} + 88 q^{18} + 188 q^{19} - 80 q^{20} + 64 q^{21} - 160 q^{22} + 352 q^{23} + 32 q^{24} + 34 q^{25} + 200 q^{26} + 44 q^{27} + 240 q^{28} + 56 q^{29} + 124 q^{30} - 60 q^{31} + 240 q^{32} - 8 q^{33} + 160 q^{34} - 384 q^{36} + 208 q^{37} - 224 q^{38} + 176 q^{39} - 312 q^{40} + 88 q^{41} - 688 q^{42} - 40 q^{43} - 736 q^{44} - 32 q^{45} - 800 q^{46} - 192 q^{47} - 520 q^{48} - 694 q^{49} - 544 q^{50} - 172 q^{51} - 112 q^{52} + 168 q^{53} + 320 q^{54} + 80 q^{55} + 448 q^{56} + 144 q^{57} + 832 q^{58} + 256 q^{59} + 512 q^{60} + 572 q^{61} + 552 q^{62} + 160 q^{63} + 544 q^{64} + 312 q^{65} + 152 q^{66} + 792 q^{67} - 336 q^{68} - 36 q^{69} - 1240 q^{70} + 256 q^{71} - 344 q^{72} - 616 q^{73} - 1576 q^{74} - 72 q^{75} - 2112 q^{76} - 928 q^{77} - 1456 q^{78} - 764 q^{79} - 392 q^{80} - 38 q^{81} - 992 q^{82} - 864 q^{83} - 1152 q^{84} - 1148 q^{85} - 160 q^{86} - 1240 q^{87} - 96 q^{88} - 696 q^{89} + 328 q^{90} - 2656 q^{91} + 880 q^{92} - 464 q^{93} + 1712 q^{94} - 2112 q^{95} + 1056 q^{96} - 56 q^{97} + 1792 q^{98} - 456 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(240))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
240.3.c $$\chi_{240}(209, \cdot)$$ 240.3.c.a 1 1
240.3.c.b 1
240.3.c.c 4
240.3.c.d 4
240.3.c.e 12
240.3.e $$\chi_{240}(31, \cdot)$$ 240.3.e.a 4 1
240.3.e.b 4
240.3.g $$\chi_{240}(151, \cdot)$$ None 0 1
240.3.i $$\chi_{240}(89, \cdot)$$ None 0 1
240.3.j $$\chi_{240}(79, \cdot)$$ 240.3.j.a 4 1
240.3.j.b 4
240.3.j.c 4
240.3.l $$\chi_{240}(161, \cdot)$$ 240.3.l.a 2 1
240.3.l.b 2
240.3.l.c 4
240.3.l.d 8
240.3.n $$\chi_{240}(41, \cdot)$$ None 0 1
240.3.p $$\chi_{240}(199, \cdot)$$ None 0 1
240.3.q $$\chi_{240}(19, \cdot)$$ 240.3.q.a 96 2
240.3.r $$\chi_{240}(101, \cdot)$$ 240.3.r.a 128 2
240.3.u $$\chi_{240}(23, \cdot)$$ None 0 2
240.3.x $$\chi_{240}(73, \cdot)$$ None 0 2
240.3.z $$\chi_{240}(83, \cdot)$$ 240.3.z.a 184 2
240.3.ba $$\chi_{240}(13, \cdot)$$ 240.3.ba.a 96 2
240.3.bd $$\chi_{240}(203, \cdot)$$ 240.3.bd.a 184 2
240.3.be $$\chi_{240}(133, \cdot)$$ 240.3.be.a 96 2
240.3.bg $$\chi_{240}(97, \cdot)$$ 240.3.bg.a 4 2
240.3.bg.b 4
240.3.bg.c 4
240.3.bg.d 4
240.3.bg.e 8
240.3.bj $$\chi_{240}(47, \cdot)$$ 240.3.bj.a 8 2
240.3.bj.b 16
240.3.bj.c 24
240.3.bm $$\chi_{240}(29, \cdot)$$ 240.3.bm.a 8 2
240.3.bm.b 176
240.3.bn $$\chi_{240}(91, \cdot)$$ 240.3.bn.a 64 2

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(240))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(240)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 20}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 16}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 1}$$