## Defining parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ 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

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

## 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

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