## Defining parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$3$$ Newforms: $$5$$ Sturm bound: $$120$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(20))$$.

Total New Old
Modular forms 58 26 32
Cusp forms 38 22 16
Eisenstein series 20 4 16

## Trace form

 $$22q + 6q^{2} - 10q^{3} - 36q^{4} + 4q^{5} + 32q^{6} + 110q^{7} + 216q^{8} - 170q^{9} + O(q^{10})$$ $$22q + 6q^{2} - 10q^{3} - 36q^{4} + 4q^{5} + 32q^{6} + 110q^{7} + 216q^{8} - 170q^{9} - 210q^{10} - 300q^{11} - 200q^{12} - 8q^{13} - 184q^{14} + 542q^{15} - 592q^{16} + 912q^{17} + 286q^{18} + 100q^{20} - 2612q^{21} + 800q^{22} - 810q^{23} + 3216q^{24} + 2066q^{25} + 324q^{26} + 2120q^{27} + 40q^{28} + 1508q^{29} - 920q^{30} - 836q^{31} - 2304q^{32} - 2780q^{33} - 4308q^{34} - 2562q^{35} - 9220q^{36} - 6388q^{37} - 3360q^{38} + 4200q^{40} + 10280q^{41} + 12120q^{42} + 3270q^{43} + 9840q^{44} + 3344q^{45} + 14792q^{46} - 2250q^{47} + 8640q^{48} - 5130q^{49} - 11730q^{50} + 1948q^{51} - 12488q^{52} + 4572q^{53} - 26768q^{54} - 6780q^{55} - 25168q^{56} - 10000q^{57} - 7428q^{58} + 11320q^{60} + 22184q^{61} + 25680q^{62} + 12950q^{63} + 33984q^{64} - 3852q^{65} + 38000q^{66} - 4810q^{67} + 2712q^{68} - 18248q^{69} - 10800q^{70} - 5988q^{71} - 36264q^{72} - 28508q^{73} - 42108q^{74} - 11282q^{75} - 36000q^{76} + 1380q^{77} - 14480q^{78} + 19120q^{80} - 1062q^{81} + 27412q^{82} + 19950q^{83} + 80672q^{84} + 36536q^{85} + 46352q^{86} + 24800q^{87} + 18080q^{88} + 60068q^{89} - 29130q^{90} - 11124q^{91} - 52680q^{92} - 22700q^{93} - 67704q^{94} - 15576q^{95} - 78208q^{96} - 7548q^{97} - 21474q^{98} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(20))$$

We only show spaces with odd 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
20.5.b $$\chi_{20}(11, \cdot)$$ 20.5.b.a 8 1
20.5.d $$\chi_{20}(19, \cdot)$$ 20.5.d.a 1 1
20.5.d.b 1
20.5.d.c 8
20.5.f $$\chi_{20}(13, \cdot)$$ 20.5.f.a 4 2

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(20))$$ into lower level spaces

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