## Defining parameters

 Level: $$N$$ = $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ Sturm bound: $$384$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(42))$$.

Total New Old
Modular forms 168 34 134
Cusp forms 120 34 86
Eisenstein series 48 0 48

## Trace form

 $$34q + 4q^{2} - 6q^{3} - 8q^{4} + 36q^{5} + 24q^{6} + 112q^{7} + 16q^{8} - 60q^{9} + O(q^{10})$$ $$34q + 4q^{2} - 6q^{3} - 8q^{4} + 36q^{5} + 24q^{6} + 112q^{7} + 16q^{8} - 60q^{9} - 48q^{10} - 108q^{11} - 24q^{12} - 64q^{13} - 32q^{14} + 288q^{15} - 32q^{16} + 264q^{17} - 132q^{18} - 424q^{19} - 48q^{20} - 696q^{21} + 48q^{22} - 252q^{23} - 96q^{24} - 242q^{25} - 304q^{26} + 108q^{27} - 32q^{28} + 36q^{29} + 600q^{30} + 944q^{31} + 64q^{32} + 1386q^{33} + 312q^{34} + 972q^{35} + 744q^{36} - 1096q^{37} - 136q^{38} - 780q^{39} - 192q^{40} - 468q^{41} - 228q^{42} + 200q^{43} - 432q^{44} - 2070q^{45} + 1008q^{46} - 168q^{47} + 192q^{48} + 4330q^{49} + 1852q^{50} + 2394q^{51} + 416q^{52} + 864q^{53} + 72q^{54} - 1692q^{55} - 224q^{56} - 1728q^{57} - 2400q^{58} - 1152q^{59} - 1368q^{60} - 6604q^{61} - 3568q^{62} - 4530q^{63} - 896q^{64} - 1128q^{65} - 2160q^{66} - 1096q^{67} + 1056q^{68} + 2664q^{69} + 816q^{70} + 2232q^{71} + 816q^{72} + 5468q^{73} + 512q^{74} + 4476q^{75} + 272q^{76} + 84q^{77} + 2664q^{78} + 3056q^{79} + 576q^{80} - 36q^{81} + 2136q^{82} - 1992q^{83} + 720q^{84} - 912q^{85} + 968q^{86} + 2592q^{87} - 480q^{88} + 1284q^{89} + 1296q^{90} + 1784q^{91} + 768q^{92} + 1698q^{93} + 1536q^{94} + 4296q^{95} - 384q^{96} + 3080q^{97} - 44q^{98} - 468q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$

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
42.4.a $$\chi_{42}(1, \cdot)$$ 42.4.a.a 1 1
42.4.a.b 1
42.4.d $$\chi_{42}(41, \cdot)$$ 42.4.d.a 8 1
42.4.e $$\chi_{42}(25, \cdot)$$ 42.4.e.a 2 2
42.4.e.b 2
42.4.e.c 4
42.4.f $$\chi_{42}(5, \cdot)$$ 42.4.f.a 16 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(42))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(42)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 2}$$