## Defining parameters

 Level: $$N$$ = $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$288$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(42))$$.

Total New Old
Modular forms 120 24 96
Cusp forms 72 24 48
Eisenstein series 48 0 48

## Trace form

 $$24q + 6q^{3} + 8q^{4} + 12q^{5} - 16q^{7} + 18q^{9} + O(q^{10})$$ $$24q + 6q^{3} + 8q^{4} + 12q^{5} - 16q^{7} + 18q^{9} - 48q^{10} - 36q^{11} - 12q^{12} - 28q^{13} - 84q^{15} - 48q^{17} - 24q^{18} - 40q^{19} - 18q^{21} + 48q^{23} + 24q^{24} + 132q^{25} + 96q^{26} + 252q^{27} + 72q^{28} + 120q^{29} + 132q^{30} + 8q^{31} - 102q^{33} - 132q^{35} - 72q^{36} + 28q^{37} + 24q^{38} - 84q^{39} + 96q^{40} - 72q^{42} + 64q^{43} - 72q^{44} - 6q^{45} - 72q^{46} - 132q^{47} - 132q^{49} - 96q^{50} - 90q^{51} - 136q^{52} - 96q^{53} - 144q^{54} - 504q^{55} - 96q^{56} - 252q^{57} - 384q^{58} - 24q^{59} - 108q^{60} - 16q^{61} + 282q^{63} - 64q^{64} + 420q^{65} + 336q^{66} + 552q^{67} + 96q^{68} + 252q^{69} + 504q^{70} + 192q^{71} + 48q^{72} + 380q^{73} + 240q^{74} + 360q^{75} + 392q^{76} + 144q^{77} + 288q^{78} + 144q^{79} - 48q^{80} - 42q^{81} + 240q^{82} - 12q^{84} + 120q^{85} - 264q^{86} - 324q^{87} - 144q^{88} - 72q^{89} - 672q^{90} - 448q^{91} - 48q^{92} - 546q^{93} - 456q^{94} - 372q^{95} - 48q^{96} - 448q^{97} - 192q^{98} + 84q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(42))$$

We only show spaces with odd 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
42.3.b $$\chi_{42}(29, \cdot)$$ 42.3.b.a 4 1
42.3.c $$\chi_{42}(13, \cdot)$$ 42.3.c.a 4 1
42.3.g $$\chi_{42}(19, \cdot)$$ 42.3.g.a 4 2
42.3.h $$\chi_{42}(11, \cdot)$$ 42.3.h.a 4 2
42.3.h.b 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(42))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(42)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 2}$$