# Properties

 Label 520.2 Level 520 Weight 2 Dimension 3976 Nonzero newspaces 32 Newform subspaces 78 Sturm bound 32256 Trace bound 6

## Defining parameters

 Level: $$N$$ = $$520 = 2^{3} \cdot 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$32$$ Newform subspaces: $$78$$ Sturm bound: $$32256$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(520))$$.

Total New Old
Modular forms 8640 4240 4400
Cusp forms 7489 3976 3513
Eisenstein series 1151 264 887

## Trace form

 $$3976 q - 16 q^{2} - 16 q^{3} - 16 q^{4} + 2 q^{5} - 56 q^{6} - 8 q^{7} - 16 q^{8} - 22 q^{9} + O(q^{10})$$ $$3976 q - 16 q^{2} - 16 q^{3} - 16 q^{4} + 2 q^{5} - 56 q^{6} - 8 q^{7} - 16 q^{8} - 22 q^{9} - 28 q^{10} - 48 q^{11} - 40 q^{12} + 2 q^{13} - 64 q^{14} - 44 q^{15} - 88 q^{16} - 30 q^{17} - 72 q^{18} - 24 q^{19} - 68 q^{20} + 32 q^{21} - 56 q^{22} - 72 q^{24} - 46 q^{25} - 80 q^{26} + 8 q^{27} - 8 q^{28} + 42 q^{29} - 36 q^{30} - 48 q^{31} + 24 q^{32} + 16 q^{33} + 48 q^{34} - 40 q^{35} - 64 q^{36} - 6 q^{37} + 24 q^{38} - 56 q^{39} - 32 q^{40} - 126 q^{41} - 120 q^{42} - 160 q^{43} - 112 q^{44} - 43 q^{45} - 248 q^{46} - 144 q^{47} - 200 q^{48} - 86 q^{49} - 176 q^{50} - 256 q^{51} - 260 q^{52} - 84 q^{53} - 288 q^{54} - 164 q^{55} - 384 q^{56} - 104 q^{57} - 304 q^{58} - 136 q^{59} - 236 q^{60} - 46 q^{61} - 256 q^{62} - 56 q^{63} - 232 q^{64} - 59 q^{65} - 272 q^{66} + 40 q^{67} - 64 q^{68} + 56 q^{69} - 96 q^{70} + 72 q^{71} - 4 q^{73} + 16 q^{74} + 140 q^{75} - 72 q^{76} + 128 q^{77} + 40 q^{78} + 160 q^{79} + 12 q^{80} - 54 q^{81} + 48 q^{82} + 216 q^{83} + 208 q^{84} + 35 q^{85} + 8 q^{86} + 24 q^{87} + 96 q^{88} - 76 q^{89} + 128 q^{90} - 200 q^{91} + 128 q^{92} - 72 q^{93} + 104 q^{94} - 108 q^{95} + 264 q^{96} - 92 q^{97} + 168 q^{98} - 400 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(520))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
520.2.a $$\chi_{520}(1, \cdot)$$ 520.2.a.a 1 1
520.2.a.b 1
520.2.a.c 2
520.2.a.d 2
520.2.a.e 2
520.2.a.f 2
520.2.a.g 2
520.2.d $$\chi_{520}(209, \cdot)$$ 520.2.d.a 2 1
520.2.d.b 6
520.2.d.c 10
520.2.e $$\chi_{520}(181, \cdot)$$ 520.2.e.a 28 1
520.2.e.b 28
520.2.f $$\chi_{520}(129, \cdot)$$ 520.2.f.a 10 1
520.2.f.b 10
520.2.g $$\chi_{520}(261, \cdot)$$ 520.2.g.a 24 1
520.2.g.b 24
520.2.j $$\chi_{520}(469, \cdot)$$ 520.2.j.a 36 1
520.2.j.b 36
520.2.k $$\chi_{520}(441, \cdot)$$ 520.2.k.a 6 1
520.2.k.b 8
520.2.p $$\chi_{520}(389, \cdot)$$ 520.2.p.a 8 1
520.2.p.b 72
520.2.q $$\chi_{520}(81, \cdot)$$ 520.2.q.a 2 2
520.2.q.b 2
520.2.q.c 2
520.2.q.d 2
520.2.q.e 2
520.2.q.f 4
520.2.q.g 6
520.2.q.h 8
520.2.s $$\chi_{520}(31, \cdot)$$ None 0 2
520.2.t $$\chi_{520}(99, \cdot)$$ 520.2.t.a 4 2
520.2.t.b 4
520.2.t.c 152
520.2.w $$\chi_{520}(57, \cdot)$$ 520.2.w.a 2 2
520.2.w.b 2
520.2.w.c 2
520.2.w.d 4
520.2.w.e 12
520.2.w.f 20
520.2.y $$\chi_{520}(317, \cdot)$$ 520.2.y.a 160 2
520.2.bb $$\chi_{520}(183, \cdot)$$ None 0 2
520.2.bc $$\chi_{520}(363, \cdot)$$ 520.2.bc.a 12 2
520.2.bc.b 12
520.2.bc.c 136
520.2.bd $$\chi_{520}(103, \cdot)$$ None 0 2
520.2.be $$\chi_{520}(27, \cdot)$$ 520.2.be.a 144 2
520.2.bh $$\chi_{520}(177, \cdot)$$ 520.2.bh.a 2 2
520.2.bh.b 2
520.2.bh.c 2
520.2.bh.d 4
520.2.bh.e 12
520.2.bh.f 20
520.2.bj $$\chi_{520}(213, \cdot)$$ 520.2.bj.a 160 2
520.2.bm $$\chi_{520}(291, \cdot)$$ 520.2.bm.a 112 2
520.2.bn $$\chi_{520}(239, \cdot)$$ None 0 2
520.2.bp $$\chi_{520}(69, \cdot)$$ 520.2.bp.a 160 2
520.2.bu $$\chi_{520}(121, \cdot)$$ 520.2.bu.a 12 2
520.2.bu.b 16
520.2.bv $$\chi_{520}(29, \cdot)$$ 520.2.bv.a 160 2
520.2.by $$\chi_{520}(61, \cdot)$$ 520.2.by.a 4 2
520.2.by.b 4
520.2.by.c 104
520.2.bz $$\chi_{520}(49, \cdot)$$ 520.2.bz.a 20 2
520.2.bz.b 20
520.2.ca $$\chi_{520}(101, \cdot)$$ 520.2.ca.a 56 2
520.2.ca.b 56
520.2.cb $$\chi_{520}(9, \cdot)$$ 520.2.cb.a 44 2
520.2.cf $$\chi_{520}(119, \cdot)$$ None 0 4
520.2.cg $$\chi_{520}(11, \cdot)$$ 520.2.cg.a 224 4
520.2.cj $$\chi_{520}(37, \cdot)$$ 520.2.cj.a 320 4
520.2.cl $$\chi_{520}(137, \cdot)$$ 520.2.cl.a 4 4
520.2.cl.b 40
520.2.cl.c 40
520.2.cm $$\chi_{520}(3, \cdot)$$ 520.2.cm.a 320 4
520.2.cn $$\chi_{520}(23, \cdot)$$ None 0 4
520.2.cs $$\chi_{520}(43, \cdot)$$ 520.2.cs.a 320 4
520.2.ct $$\chi_{520}(87, \cdot)$$ None 0 4
520.2.cu $$\chi_{520}(197, \cdot)$$ 520.2.cu.a 320 4
520.2.cw $$\chi_{520}(33, \cdot)$$ 520.2.cw.a 4 4
520.2.cw.b 40
520.2.cw.c 40
520.2.cz $$\chi_{520}(19, \cdot)$$ 520.2.cz.a 8 4
520.2.cz.b 8
520.2.cz.c 304
520.2.da $$\chi_{520}(71, \cdot)$$ None 0 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(520))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(520)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(130))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(260))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(520))$$$$^{\oplus 1}$$