# Properties

 Label 80.2 Level 80 Weight 2 Dimension 86 Nonzero newspaces 7 Newform subspaces 13 Sturm bound 768 Trace bound 3

## 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

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

## 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 available 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

$$S_{2}^{\mathrm{old}}(\Gamma_1(80)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 1}$$