# Properties

 Label 420.2 Level 420 Weight 2 Dimension 1708 Nonzero newspaces 24 Newform subspaces 57 Sturm bound 18432 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Newform subspaces: $$57$$ Sturm bound: $$18432$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(420))$$.

Total New Old
Modular forms 5088 1836 3252
Cusp forms 4129 1708 2421
Eisenstein series 959 128 831

## Trace form

 $$1708 q - 6 q^{3} + 4 q^{4} - 10 q^{5} + 4 q^{6} - 12 q^{7} + 36 q^{8} + 18 q^{9} + O(q^{10})$$ $$1708 q - 6 q^{3} + 4 q^{4} - 10 q^{5} + 4 q^{6} - 12 q^{7} + 36 q^{8} + 18 q^{9} + 40 q^{10} + 28 q^{11} + 32 q^{12} + 68 q^{13} + 48 q^{14} + 46 q^{15} + 12 q^{16} + 64 q^{17} - 8 q^{18} + 32 q^{19} - 40 q^{20} + 34 q^{21} - 88 q^{22} + 24 q^{23} - 72 q^{24} + 46 q^{25} - 92 q^{26} + 12 q^{27} - 132 q^{28} + 96 q^{29} - 114 q^{30} + 8 q^{31} - 100 q^{32} + 6 q^{33} - 144 q^{34} + 50 q^{35} - 108 q^{36} + 52 q^{37} - 68 q^{38} - 16 q^{39} - 80 q^{40} + 8 q^{41} - 116 q^{42} - 8 q^{43} - 48 q^{44} - 23 q^{45} - 48 q^{46} - 12 q^{47} - 80 q^{48} - 80 q^{49} + 20 q^{50} - 18 q^{51} - 88 q^{52} - 56 q^{53} + 4 q^{54} - 32 q^{55} - 68 q^{56} - 116 q^{57} - 104 q^{58} - 56 q^{59} - 16 q^{60} - 88 q^{61} - 104 q^{62} + 2 q^{63} - 140 q^{64} - 178 q^{65} - 4 q^{66} - 64 q^{67} - 160 q^{68} - 184 q^{69} - 304 q^{70} - 96 q^{71} - 48 q^{72} - 220 q^{73} - 132 q^{74} - 105 q^{75} - 232 q^{76} - 192 q^{77} + 40 q^{78} - 216 q^{79} - 152 q^{80} - 246 q^{81} - 208 q^{82} - 72 q^{83} + 64 q^{84} - 268 q^{85} - 92 q^{86} - 124 q^{87} - 184 q^{88} - 64 q^{89} - 12 q^{90} - 64 q^{91} + 56 q^{92} - 166 q^{93} - 184 q^{94} + 18 q^{95} - 20 q^{96} - 96 q^{97} + 60 q^{98} - 4 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(420))$$

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
420.2.a $$\chi_{420}(1, \cdot)$$ 420.2.a.a 1 1
420.2.a.b 1
420.2.a.c 1
420.2.a.d 1
420.2.c $$\chi_{420}(391, \cdot)$$ 420.2.c.a 16 1
420.2.c.b 16
420.2.d $$\chi_{420}(41, \cdot)$$ 420.2.d.a 2 1
420.2.d.b 2
420.2.d.c 4
420.2.d.d 4
420.2.f $$\chi_{420}(209, \cdot)$$ 420.2.f.a 8 1
420.2.f.b 8
420.2.i $$\chi_{420}(139, \cdot)$$ 420.2.i.a 48 1
420.2.k $$\chi_{420}(169, \cdot)$$ 420.2.k.a 2 1
420.2.k.b 2
420.2.l $$\chi_{420}(239, \cdot)$$ 420.2.l.a 4 1
420.2.l.b 4
420.2.l.c 8
420.2.l.d 8
420.2.l.e 8
420.2.l.f 8
420.2.l.g 16
420.2.l.h 16
420.2.n $$\chi_{420}(71, \cdot)$$ 420.2.n.a 24 1
420.2.n.b 24
420.2.q $$\chi_{420}(121, \cdot)$$ 420.2.q.a 2 2
420.2.q.b 2
420.2.q.c 4
420.2.q.d 4
420.2.s $$\chi_{420}(113, \cdot)$$ 420.2.s.a 24 2
420.2.t $$\chi_{420}(43, \cdot)$$ 420.2.t.a 4 2
420.2.t.b 4
420.2.t.c 32
420.2.t.d 32
420.2.w $$\chi_{420}(83, \cdot)$$ 420.2.w.a 8 2
420.2.w.b 8
420.2.w.c 160
420.2.x $$\chi_{420}(13, \cdot)$$ 420.2.x.a 16 2
420.2.ba $$\chi_{420}(179, \cdot)$$ 420.2.ba.a 8 2
420.2.ba.b 8
420.2.ba.c 160
420.2.bb $$\chi_{420}(109, \cdot)$$ 420.2.bb.a 16 2
420.2.bf $$\chi_{420}(11, \cdot)$$ 420.2.bf.a 128 2
420.2.bh $$\chi_{420}(101, \cdot)$$ 420.2.bh.a 10 2
420.2.bh.b 10
420.2.bi $$\chi_{420}(31, \cdot)$$ 420.2.bi.a 4 2
420.2.bi.b 4
420.2.bi.c 28
420.2.bi.d 28
420.2.bk $$\chi_{420}(19, \cdot)$$ 420.2.bk.a 96 2
420.2.bn $$\chi_{420}(89, \cdot)$$ 420.2.bn.a 32 2
420.2.bo $$\chi_{420}(73, \cdot)$$ 420.2.bo.a 32 4
420.2.br $$\chi_{420}(47, \cdot)$$ 420.2.br.a 352 4
420.2.bs $$\chi_{420}(67, \cdot)$$ 420.2.bs.a 192 4
420.2.bv $$\chi_{420}(53, \cdot)$$ 420.2.bv.a 8 4
420.2.bv.b 8
420.2.bv.c 48

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(420))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(420)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 2}$$