## Defining parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$72$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(20))$$.

Total New Old
Modular forms 34 14 20
Cusp forms 14 10 4
Eisenstein series 20 4 16

## Trace form

 $$10q - 2q^{2} + 2q^{3} - 4q^{4} - 10q^{5} - 16q^{6} - 14q^{7} - 8q^{8} - 8q^{9} + O(q^{10})$$ $$10q - 2q^{2} + 2q^{3} - 4q^{4} - 10q^{5} - 16q^{6} - 14q^{7} - 8q^{8} - 8q^{9} + 6q^{10} + 20q^{11} + 40q^{12} + 2q^{13} + 56q^{14} + 2q^{15} + 48q^{16} - 22q^{17} + 22q^{18} - 44q^{20} - 20q^{21} - 80q^{22} - 46q^{23} - 144q^{24} + 42q^{25} - 108q^{26} + 32q^{27} - 40q^{28} + 32q^{29} + 40q^{30} - 28q^{31} + 128q^{32} + 100q^{33} + 156q^{34} + 98q^{35} + 92q^{36} + 82q^{37} + 80q^{38} - 24q^{40} - 52q^{41} - 120q^{42} - 30q^{43} - 80q^{44} - 220q^{45} - 136q^{46} - 78q^{47} - 168q^{49} + 86q^{50} + 4q^{51} + 56q^{52} - 190q^{53} + 112q^{54} - 60q^{55} + 144q^{56} - 16q^{57} - 36q^{58} + 40q^{60} + 140q^{61} - 80q^{62} + 98q^{63} - 64q^{64} + 250q^{65} + 80q^{66} - 14q^{67} - 136q^{68} + 472q^{69} - 80q^{70} + 196q^{71} - 72q^{72} + 362q^{73} - 204q^{74} - 62q^{75} + 100q^{77} - 80q^{78} - 144q^{80} - 366q^{81} + 116q^{82} - 126q^{83} + 32q^{84} - 302q^{85} + 224q^{86} - 16q^{87} + 160q^{88} - 528q^{89} - 114q^{90} - 252q^{91} - 120q^{92} - 428q^{93} - 104q^{94} + 64q^{95} - 256q^{96} - 198q^{97} + 102q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$

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
20.3.b $$\chi_{20}(11, \cdot)$$ 20.3.b.a 4 1
20.3.d $$\chi_{20}(19, \cdot)$$ 20.3.d.a 1 1
20.3.d.b 1
20.3.d.c 2
20.3.f $$\chi_{20}(13, \cdot)$$ 20.3.f.a 2 2

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(20))$$ into lower level spaces

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