## Defining parameters

 Level: $$N$$ = $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$68$$ Sturm bound: $$12960$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(342))$$.

Total New Old
Modular forms 3528 851 2677
Cusp forms 2953 851 2102
Eisenstein series 575 0 575

## Trace form

 $$851 q + 2 q^{2} + 6 q^{3} + 2 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{8} - 6 q^{9} + O(q^{10})$$ $$851 q + 2 q^{2} + 6 q^{3} + 2 q^{4} - 6 q^{6} + 4 q^{7} - 4 q^{8} - 6 q^{9} - 6 q^{11} + 28 q^{13} + 22 q^{14} + 2 q^{16} + 30 q^{17} + 12 q^{18} + 44 q^{19} + 18 q^{20} - 12 q^{21} + 21 q^{22} + 6 q^{23} + 6 q^{24} + 26 q^{25} + 10 q^{26} - 2 q^{28} + 30 q^{29} + 10 q^{31} + 2 q^{32} + 18 q^{33} - 6 q^{34} + 36 q^{35} - 6 q^{36} + 34 q^{37} - q^{38} + 36 q^{41} + 16 q^{43} - 15 q^{44} - 108 q^{45} - 84 q^{46} - 156 q^{47} - 24 q^{48} - 114 q^{49} - 226 q^{50} - 198 q^{51} - 14 q^{52} - 264 q^{53} - 126 q^{54} - 216 q^{55} - 104 q^{56} - 183 q^{57} - 96 q^{58} - 264 q^{59} - 72 q^{60} - 194 q^{61} - 146 q^{62} - 120 q^{63} - 22 q^{64} - 252 q^{65} - 144 q^{66} - 80 q^{67} - 33 q^{68} - 108 q^{69} - 108 q^{70} - 96 q^{71} - 24 q^{72} - 35 q^{73} - 8 q^{74} + 30 q^{75} - q^{76} + 24 q^{77} + 12 q^{78} + 70 q^{79} + 18 q^{81} + 96 q^{83} + 12 q^{84} + 72 q^{85} + 34 q^{86} - 36 q^{87} - 6 q^{88} + 48 q^{89} + 80 q^{91} + 24 q^{92} + 60 q^{94} - 54 q^{95} - 44 q^{97} + 84 q^{98} - 162 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(342))$$

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
342.2.a $$\chi_{342}(1, \cdot)$$ 342.2.a.a 1 1
342.2.a.b 1
342.2.a.c 1
342.2.a.d 1
342.2.a.e 1
342.2.a.f 1
342.2.a.g 1
342.2.b $$\chi_{342}(341, \cdot)$$ 342.2.b.a 2 1
342.2.b.b 2
342.2.e $$\chi_{342}(115, \cdot)$$ 342.2.e.a 6 2
342.2.e.b 6
342.2.e.c 12
342.2.e.d 12
342.2.f $$\chi_{342}(7, \cdot)$$ 342.2.f.a 2 2
342.2.f.b 2
342.2.f.c 2
342.2.f.d 2
342.2.f.e 4
342.2.f.f 10
342.2.f.g 18
342.2.g $$\chi_{342}(163, \cdot)$$ 342.2.g.a 2 2
342.2.g.b 2
342.2.g.c 2
342.2.g.d 2
342.2.g.e 2
342.2.g.f 4
342.2.h $$\chi_{342}(121, \cdot)$$ 342.2.h.a 2 2
342.2.h.b 2
342.2.h.c 2
342.2.h.d 2
342.2.h.e 4
342.2.h.f 10
342.2.h.g 18
342.2.j $$\chi_{342}(65, \cdot)$$ 342.2.j.a 2 2
342.2.j.b 2
342.2.j.c 2
342.2.j.d 8
342.2.j.e 8
342.2.j.f 18
342.2.n $$\chi_{342}(293, \cdot)$$ 342.2.n.a 2 2
342.2.n.b 2
342.2.n.c 2
342.2.n.d 8
342.2.n.e 8
342.2.n.f 18
342.2.p $$\chi_{342}(113, \cdot)$$ 342.2.p.a 20 2
342.2.p.b 20
342.2.s $$\chi_{342}(107, \cdot)$$ 342.2.s.a 4 2
342.2.s.b 4
342.2.u $$\chi_{342}(55, \cdot)$$ 342.2.u.a 6 6
342.2.u.b 6
342.2.u.c 6
342.2.u.d 6
342.2.u.e 6
342.2.u.f 12
342.2.u.g 12
342.2.v $$\chi_{342}(25, \cdot)$$ 342.2.v.a 54 6
342.2.v.b 66
342.2.w $$\chi_{342}(43, \cdot)$$ 342.2.w.a 54 6
342.2.w.b 66
342.2.x $$\chi_{342}(29, \cdot)$$ 342.2.x.a 12 6
342.2.x.b 48
342.2.x.c 60
342.2.bb $$\chi_{342}(53, \cdot)$$ 342.2.bb.a 24 6
342.2.bb.b 24
342.2.bf $$\chi_{342}(155, \cdot)$$ 342.2.bf.a 12 6
342.2.bf.b 48
342.2.bf.c 60

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(342))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(342)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(171))$$$$^{\oplus 2}$$