## Defining parameters

 Level: $$N$$ = $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$192$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(42))$$.

Total New Old
Modular forms 72 13 59
Cusp forms 25 13 12
Eisenstein series 47 0 47

## Trace form

 $$13 q + q^{2} - q^{3} - 3 q^{4} - 6 q^{5} - 5 q^{6} - 9 q^{7} + q^{8} - 7 q^{9} + O(q^{10})$$ $$13 q + q^{2} - q^{3} - 3 q^{4} - 6 q^{5} - 5 q^{6} - 9 q^{7} + q^{8} - 7 q^{9} - 6 q^{10} - 12 q^{11} - q^{12} - 2 q^{13} + 7 q^{14} + 6 q^{15} + q^{16} + 6 q^{17} + 13 q^{18} + 4 q^{19} + 6 q^{20} + 17 q^{21} + 12 q^{22} + 12 q^{23} + 7 q^{24} + 7 q^{25} + 2 q^{26} - q^{27} - 5 q^{28} + 6 q^{29} - 6 q^{30} - 8 q^{31} + q^{32} - 12 q^{33} - 6 q^{34} - 6 q^{35} - 7 q^{36} - 18 q^{37} - 16 q^{38} - 14 q^{39} - 6 q^{40} - 6 q^{41} - 23 q^{42} - 36 q^{43} - 12 q^{44} - 6 q^{45} - 24 q^{46} + 12 q^{47} - q^{48} + q^{49} - 17 q^{50} + 30 q^{51} + 22 q^{52} + 18 q^{53} + 19 q^{54} + 36 q^{55} - 5 q^{56} - 8 q^{57} + 42 q^{58} + 12 q^{59} + 18 q^{60} + 22 q^{61} + 8 q^{62} + 23 q^{63} - 3 q^{64} + 12 q^{66} + 44 q^{67} + 6 q^{68} + 42 q^{70} - 11 q^{72} + 22 q^{73} + 2 q^{74} - 7 q^{75} + 4 q^{76} - 24 q^{77} - 10 q^{78} - 40 q^{79} - 6 q^{80} - 55 q^{81} - 30 q^{82} - 36 q^{83} - 19 q^{84} - 84 q^{85} - 16 q^{86} - 30 q^{87} - 6 q^{89} - 6 q^{90} - 14 q^{91} + 4 q^{93} - 24 q^{94} + 12 q^{95} + 7 q^{96} - 2 q^{97} + 25 q^{98} + 12 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$

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
42.2.a $$\chi_{42}(1, \cdot)$$ 42.2.a.a 1 1
42.2.d $$\chi_{42}(41, \cdot)$$ 42.2.d.a 4 1
42.2.e $$\chi_{42}(25, \cdot)$$ 42.2.e.a 2 2
42.2.e.b 2
42.2.f $$\chi_{42}(5, \cdot)$$ 42.2.f.a 4 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(42))$$ into lower level spaces

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