## 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

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

## 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

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