## Defining parameters

 Level: $$N$$ = $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$16$$ Newform subspaces: $$29$$ Sturm bound: $$19440$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(342))$$.

Total New Old
Modular forms 6768 1698 5070
Cusp forms 6192 1698 4494
Eisenstein series 576 0 576

## Trace form

 $$1698q + 36q^{5} + 24q^{6} + 12q^{7} - 24q^{9} + O(q^{10})$$ $$1698q + 36q^{5} + 24q^{6} + 12q^{7} - 24q^{9} - 24q^{10} - 36q^{11} - 24q^{12} - 132q^{13} - 144q^{14} - 36q^{15} - 18q^{17} + 48q^{18} + 114q^{19} + 108q^{20} + 84q^{21} + 228q^{22} + 54q^{23} + 216q^{25} + 144q^{26} + 12q^{28} - 180q^{29} - 144q^{30} - 360q^{31} - 108q^{33} - 120q^{34} - 72q^{35} - 24q^{36} - 120q^{37} + 72q^{38} + 204q^{39} + 48q^{40} + 324q^{41} + 96q^{42} + 702q^{43} + 324q^{44} + 1296q^{45} + 1032q^{46} + 1530q^{47} + 96q^{48} + 1254q^{49} + 1296q^{50} + 1044q^{51} + 96q^{52} + 648q^{53} + 144q^{54} + 144q^{55} - 144q^{56} - 324q^{57} - 240q^{58} - 1062q^{59} - 216q^{60} - 1488q^{61} - 972q^{62} - 1452q^{63} - 48q^{64} - 2790q^{65} - 1296q^{66} - 1494q^{67} - 468q^{68} - 1440q^{69} - 1200q^{70} - 2502q^{71} - 96q^{72} + 120q^{73} - 144q^{74} + 24q^{75} - 24q^{76} - 90q^{77} - 288q^{78} - 96q^{79} + 504q^{81} - 456q^{82} + 306q^{83} + 288q^{84} - 648q^{85} - 72q^{86} + 108q^{87} - 240q^{88} - 18q^{89} - 528q^{91} - 144q^{92} - 444q^{93} - 648q^{94} + 1818q^{95} - 96q^{96} + 1158q^{97} + 288q^{98} + 1512q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(342))$$

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
342.3.c $$\chi_{342}(305, \cdot)$$ 342.3.c.a 4 1
342.3.c.b 8
342.3.d $$\chi_{342}(37, \cdot)$$ 342.3.d.a 2 1
342.3.d.b 8
342.3.d.c 8
342.3.i $$\chi_{342}(11, \cdot)$$ 342.3.i.a 80 2
342.3.k $$\chi_{342}(103, \cdot)$$ 342.3.k.a 80 2
342.3.l $$\chi_{342}(151, \cdot)$$ 342.3.l.a 80 2
342.3.m $$\chi_{342}(145, \cdot)$$ 342.3.m.a 4 2
342.3.m.b 8
342.3.m.c 8
342.3.m.d 16
342.3.o $$\chi_{342}(77, \cdot)$$ 342.3.o.a 72 2
342.3.q $$\chi_{342}(83, \cdot)$$ 342.3.q.a 80 2
342.3.r $$\chi_{342}(125, \cdot)$$ 342.3.r.a 4 2
342.3.r.b 4
342.3.r.c 12
342.3.r.d 12
342.3.t $$\chi_{342}(31, \cdot)$$ 342.3.t.a 80 2
342.3.y $$\chi_{342}(5, \cdot)$$ 342.3.y.a 240 6
342.3.z $$\chi_{342}(91, \cdot)$$ 342.3.z.a 12 6
342.3.z.b 24
342.3.z.c 24
342.3.z.d 36
342.3.ba $$\chi_{342}(17, \cdot)$$ 342.3.ba.a 36 6
342.3.ba.b 36
342.3.bc $$\chi_{342}(193, \cdot)$$ 342.3.bc.a 240 6
342.3.bd $$\chi_{342}(13, \cdot)$$ 342.3.bd.a 240 6
342.3.be $$\chi_{342}(23, \cdot)$$ 342.3.be.a 240 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(342))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(342)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(171))$$$$^{\oplus 2}$$