## Defining parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newforms: $$4$$ Sturm bound: $$96$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(20))$$.

Total New Old
Modular forms 46 25 21
Cusp forms 26 17 9
Eisenstein series 20 8 12

## Trace form

 $$17q - 2q^{2} + 4q^{3} + 15q^{5} + 8q^{6} - 16q^{7} - 44q^{8} - 109q^{9} + O(q^{10})$$ $$17q - 2q^{2} + 4q^{3} + 15q^{5} + 8q^{6} - 16q^{7} - 44q^{8} - 109q^{9} - 74q^{10} - 20q^{11} - 80q^{12} + 128q^{13} + 172q^{15} + 184q^{16} - 116q^{17} + 306q^{18} - 124q^{19} + 316q^{20} - 56q^{21} + 360q^{22} + 48q^{23} + 77q^{25} - 460q^{26} - 152q^{27} - 880q^{28} - 174q^{29} - 1240q^{30} - 272q^{31} - 632q^{32} - 160q^{33} - 232q^{35} + 892q^{36} + 580q^{37} + 1600q^{38} + 1256q^{39} + 1836q^{40} + 1006q^{41} + 1160q^{42} - 100q^{43} - 155q^{45} - 1432q^{46} + 168q^{47} - 2720q^{48} + 447q^{49} - 2214q^{50} - 1144q^{51} - 1524q^{52} - 1196q^{53} - 20q^{55} + 2048q^{56} - 784q^{57} + 2712q^{58} - 308q^{59} + 3280q^{60} - 1526q^{61} + 2440q^{62} + 176q^{63} - 2260q^{65} - 1680q^{66} - 1036q^{67} - 2428q^{68} - 872q^{69} - 3040q^{70} + 984q^{71} - 2172q^{72} + 2448q^{73} + 2228q^{75} + 800q^{76} + 4080q^{77} + 3720q^{78} + 368q^{79} + 2096q^{80} + 4917q^{81} + 536q^{82} + 948q^{83} + 3588q^{85} - 2552q^{86} - 744q^{87} - 2400q^{88} - 4066q^{89} - 2154q^{90} - 2288q^{91} - 1840q^{92} - 2576q^{93} - 956q^{95} + 1088q^{96} - 6760q^{97} + 326q^{98} - 1300q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$

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
20.4.a $$\chi_{20}(1, \cdot)$$ 20.4.a.a 1 1
20.4.c $$\chi_{20}(9, \cdot)$$ 20.4.c.a 2 1
20.4.e $$\chi_{20}(3, \cdot)$$ 20.4.e.a 2 2
20.4.e.b 12

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(20))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(20)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 2}$$