## Defining parameters

 Level: $$N$$ = $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newforms: $$5$$ 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

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

## 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

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