## Defining parameters

 Level: $$N$$ = $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$14$$ Newform subspaces: $$42$$ Sturm bound: $$12288$$ Trace bound: $$12$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(320))$$.

Total New Old
Modular forms 3360 1650 1710
Cusp forms 2785 1518 1267
Eisenstein series 575 132 443

## Trace form

 $$1518q - 16q^{2} - 12q^{3} - 16q^{4} - 24q^{5} - 48q^{6} - 8q^{7} - 16q^{8} - 14q^{9} + O(q^{10})$$ $$1518q - 16q^{2} - 12q^{3} - 16q^{4} - 24q^{5} - 48q^{6} - 8q^{7} - 16q^{8} - 14q^{9} - 24q^{10} - 28q^{11} - 16q^{12} - 16q^{14} - 12q^{15} - 48q^{16} - 12q^{17} - 16q^{18} + 4q^{19} - 24q^{20} - 56q^{21} - 32q^{22} - 8q^{23} - 96q^{24} - 46q^{25} - 128q^{26} - 24q^{27} - 96q^{28} - 48q^{29} - 104q^{30} - 80q^{31} - 96q^{32} - 64q^{33} - 96q^{34} - 20q^{35} - 208q^{36} - 32q^{37} - 96q^{38} - 16q^{39} - 64q^{40} - 60q^{41} - 96q^{42} + 12q^{43} - 32q^{44} - 20q^{45} - 48q^{46} + 24q^{47} - 16q^{48} - 10q^{49} - 88q^{51} + 80q^{52} + 32q^{53} + 112q^{54} - 80q^{55} + 64q^{56} - 32q^{57} + 128q^{58} - 148q^{59} + 72q^{60} - 32q^{61} + 48q^{62} - 200q^{63} + 176q^{64} - 96q^{65} + 112q^{66} - 236q^{67} + 80q^{68} - 88q^{69} + 72q^{70} - 224q^{71} + 128q^{72} - 84q^{73} + 96q^{74} - 176q^{75} + 80q^{76} - 120q^{77} + 32q^{78} - 144q^{79} - 64q^{80} - 218q^{81} - 176q^{82} - 92q^{83} - 240q^{84} - 128q^{85} - 256q^{86} - 120q^{87} - 176q^{88} - 180q^{89} - 168q^{90} - 88q^{91} - 320q^{92} - 192q^{93} - 208q^{94} - 84q^{95} - 320q^{96} - 124q^{97} - 288q^{98} - 132q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(320))$$

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
320.2.a $$\chi_{320}(1, \cdot)$$ 320.2.a.a 1 1
320.2.a.b 1
320.2.a.c 1
320.2.a.d 1
320.2.a.e 1
320.2.a.f 1
320.2.a.g 2
320.2.c $$\chi_{320}(129, \cdot)$$ 320.2.c.a 2 1
320.2.c.b 2
320.2.c.c 2
320.2.c.d 4
320.2.d $$\chi_{320}(161, \cdot)$$ 320.2.d.a 4 1
320.2.d.b 4
320.2.f $$\chi_{320}(289, \cdot)$$ 320.2.f.a 4 1
320.2.f.b 8
320.2.j $$\chi_{320}(47, \cdot)$$ 320.2.j.a 2 2
320.2.j.b 18
320.2.l $$\chi_{320}(81, \cdot)$$ 320.2.l.a 16 2
320.2.n $$\chi_{320}(63, \cdot)$$ 320.2.n.a 2 2
320.2.n.b 2
320.2.n.c 2
320.2.n.d 2
320.2.n.e 2
320.2.n.f 2
320.2.n.g 2
320.2.n.h 2
320.2.n.i 4
320.2.o $$\chi_{320}(223, \cdot)$$ 320.2.o.a 2 2
320.2.o.b 2
320.2.o.c 2
320.2.o.d 2
320.2.o.e 8
320.2.o.f 8
320.2.q $$\chi_{320}(49, \cdot)$$ 320.2.q.a 2 2
320.2.q.b 2
320.2.q.c 16
320.2.s $$\chi_{320}(207, \cdot)$$ 320.2.s.a 2 2
320.2.s.b 18
320.2.u $$\chi_{320}(87, \cdot)$$ None 0 4
320.2.x $$\chi_{320}(41, \cdot)$$ None 0 4
320.2.z $$\chi_{320}(9, \cdot)$$ None 0 4
320.2.ba $$\chi_{320}(7, \cdot)$$ None 0 4
320.2.bd $$\chi_{320}(43, \cdot)$$ 320.2.bd.a 368 8
320.2.be $$\chi_{320}(21, \cdot)$$ 320.2.be.a 256 8
320.2.bf $$\chi_{320}(29, \cdot)$$ 320.2.bf.a 368 8
320.2.bj $$\chi_{320}(3, \cdot)$$ 320.2.bj.a 368 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(320))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(320)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 2}$$