## 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

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

## 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

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