## Defining parameters

 Level: $$N$$ = $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$7$$ Newform subspaces: $$13$$ Sturm bound: $$768$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(80))$$.

Total New Old
Modular forms 248 112 136
Cusp forms 137 86 51
Eisenstein series 111 26 85

## Trace form

 $$86q - 4q^{2} - 2q^{3} - 8q^{4} - 8q^{5} - 24q^{6} - 6q^{7} - 16q^{8} - 4q^{9} + O(q^{10})$$ $$86q - 4q^{2} - 2q^{3} - 8q^{4} - 8q^{5} - 24q^{6} - 6q^{7} - 16q^{8} - 4q^{9} - 8q^{10} - 16q^{11} - 10q^{13} - 22q^{15} - 18q^{17} - 12q^{18} - 28q^{19} - 12q^{20} - 44q^{21} - 8q^{22} - 18q^{23} - 8q^{24} - 10q^{25} - 24q^{26} + 4q^{27} + 8q^{28} - 20q^{29} + 28q^{30} + 28q^{31} + 16q^{32} + 16q^{33} + 48q^{34} + 10q^{35} + 56q^{36} + 18q^{37} + 72q^{38} + 20q^{39} + 80q^{40} + 28q^{41} + 72q^{42} - 10q^{43} + 56q^{44} + 18q^{45} + 24q^{46} - 38q^{47} + 40q^{48} - 16q^{49} + 48q^{50} - 36q^{51} + 32q^{52} - 6q^{53} + 40q^{54} - 20q^{55} - 40q^{57} + 16q^{58} - 28q^{59} - 40q^{60} - 40q^{62} + 26q^{63} - 32q^{64} - 50q^{65} - 88q^{66} + 50q^{67} - 64q^{68} + 12q^{69} - 88q^{70} + 44q^{71} - 136q^{72} - 22q^{73} - 96q^{74} + 106q^{75} - 104q^{76} - 8q^{77} - 136q^{78} + 88q^{79} - 96q^{80} - 58q^{81} - 72q^{82} + 86q^{83} - 144q^{84} - 10q^{85} - 80q^{86} + 116q^{87} - 112q^{88} - 24q^{89} - 124q^{90} + 12q^{91} - 56q^{93} - 32q^{94} + 52q^{95} - 48q^{96} - 38q^{97} + 36q^{98} + 72q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$

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
80.2.a $$\chi_{80}(1, \cdot)$$ 80.2.a.a 1 1
80.2.a.b 1
80.2.c $$\chi_{80}(49, \cdot)$$ 80.2.c.a 2 1
80.2.d $$\chi_{80}(41, \cdot)$$ None 0 1
80.2.f $$\chi_{80}(9, \cdot)$$ None 0 1
80.2.j $$\chi_{80}(43, \cdot)$$ 80.2.j.a 2 2
80.2.j.b 18
80.2.l $$\chi_{80}(21, \cdot)$$ 80.2.l.a 16 2
80.2.n $$\chi_{80}(47, \cdot)$$ 80.2.n.a 2 2
80.2.n.b 4
80.2.o $$\chi_{80}(7, \cdot)$$ None 0 2
80.2.q $$\chi_{80}(29, \cdot)$$ 80.2.q.a 2 2
80.2.q.b 2
80.2.q.c 16
80.2.s $$\chi_{80}(3, \cdot)$$ 80.2.s.a 2 2
80.2.s.b 18

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(80))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(80)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 2}$$