## Defining parameters

 Level: $$N$$ = $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$5$$ Newform subspaces: $$7$$ Sturm bound: $$384$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(40))$$.

Total New Old
Modular forms 168 79 89
Cusp forms 120 67 53
Eisenstein series 48 12 36

## Trace form

 $$67 q + 4 q^{3} + 20 q^{4} - 9 q^{5} - 64 q^{6} - 8 q^{7} - 84 q^{8} + 63 q^{9} + O(q^{10})$$ $$67 q + 4 q^{3} + 20 q^{4} - 9 q^{5} - 64 q^{6} - 8 q^{7} - 84 q^{8} + 63 q^{9} + 52 q^{10} + 108 q^{11} + 216 q^{12} + 10 q^{13} + 96 q^{14} - 196 q^{15} - 152 q^{16} - 138 q^{17} - 484 q^{18} - 316 q^{19} - 496 q^{20} - 184 q^{21} - 96 q^{22} + 528 q^{23} + 248 q^{24} + 67 q^{25} - 200 q^{26} + 520 q^{27} + 184 q^{28} + 378 q^{29} - 208 q^{30} + 160 q^{31} + 360 q^{32} - 456 q^{33} + 228 q^{34} - 424 q^{35} + 832 q^{36} - 494 q^{37} + 1456 q^{38} - 936 q^{39} + 1332 q^{40} - 426 q^{41} + 1904 q^{42} - 916 q^{43} + 448 q^{44} + 307 q^{45} + 272 q^{46} - 488 q^{47} - 360 q^{48} + 2307 q^{49} + 92 q^{50} + 3320 q^{51} - 876 q^{52} + 1042 q^{53} - 3832 q^{54} - 604 q^{55} - 2000 q^{56} - 2304 q^{57} - 1060 q^{58} - 1444 q^{59} - 3128 q^{60} - 1966 q^{61} - 2288 q^{62} - 3096 q^{63} - 4960 q^{64} + 58 q^{65} - 3920 q^{66} - 1292 q^{67} - 1148 q^{68} + 1400 q^{69} + 1192 q^{70} + 1336 q^{71} + 5444 q^{72} + 742 q^{73} + 3804 q^{74} + 3660 q^{75} + 3568 q^{76} + 320 q^{77} + 5560 q^{78} + 5648 q^{79} + 5000 q^{80} + 483 q^{81} + 5916 q^{82} + 4316 q^{83} + 8352 q^{84} + 982 q^{85} + 9808 q^{86} + 1696 q^{87} + 1216 q^{88} + 142 q^{89} - 2460 q^{90} - 3248 q^{91} - 8400 q^{92} - 1520 q^{93} - 6744 q^{94} - 4020 q^{95} - 12944 q^{96} - 1650 q^{97} - 11080 q^{98} - 4572 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$

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
40.4.a $$\chi_{40}(1, \cdot)$$ 40.4.a.a 1 1
40.4.a.b 1
40.4.a.c 1
40.4.c $$\chi_{40}(9, \cdot)$$ 40.4.c.a 4 1
40.4.d $$\chi_{40}(21, \cdot)$$ 40.4.d.a 12 1
40.4.f $$\chi_{40}(29, \cdot)$$ 40.4.f.a 16 1
40.4.j $$\chi_{40}(7, \cdot)$$ None 0 2
40.4.k $$\chi_{40}(3, \cdot)$$ 40.4.k.a 32 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(40))$$ into lower level spaces

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