# Properties

 Label 342.4 Level 342 Weight 4 Dimension 2551 Nonzero newspaces 16 Sturm bound 25920 Trace bound 4

# Learn more

## Defining parameters

 Level: $$N$$ = $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$25920$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(342))$$.

Total New Old
Modular forms 10008 2551 7457
Cusp forms 9432 2551 6881
Eisenstein series 576 0 576

## Trace form

 $$2551 q - 8 q^{2} - 6 q^{3} + 16 q^{4} - 24 q^{5} + 36 q^{6} + 8 q^{7} + 16 q^{8} + 210 q^{9} + O(q^{10})$$ $$2551 q - 8 q^{2} - 6 q^{3} + 16 q^{4} - 24 q^{5} + 36 q^{6} + 8 q^{7} + 16 q^{8} + 210 q^{9} + 24 q^{10} - 54 q^{11} - 48 q^{12} + 260 q^{13} - 100 q^{14} - 180 q^{15} + 64 q^{16} - 384 q^{17} - 312 q^{18} - 986 q^{19} - 384 q^{20} + 72 q^{21} - 78 q^{22} + 900 q^{23} + 48 q^{24} + 1480 q^{25} + 1316 q^{26} + 752 q^{28} + 1386 q^{29} + 576 q^{30} + 278 q^{31} - 128 q^{32} - 1530 q^{33} - 684 q^{34} - 3972 q^{35} - 1032 q^{36} - 1558 q^{37} - 362 q^{38} + 1164 q^{39} + 96 q^{40} + 48 q^{41} + 1632 q^{42} - 2674 q^{43} - 1980 q^{44} - 6804 q^{45} - 6168 q^{46} - 3426 q^{47} + 3054 q^{49} + 2152 q^{50} + 4986 q^{51} + 536 q^{52} + 8844 q^{53} + 3492 q^{54} + 14328 q^{55} + 5504 q^{56} + 9093 q^{57} + 5448 q^{58} + 12834 q^{59} + 5040 q^{60} + 7946 q^{61} + 7892 q^{62} + 5604 q^{63} - 896 q^{64} - 4062 q^{65} - 5472 q^{66} - 11854 q^{67} - 3372 q^{68} - 13788 q^{69} - 14424 q^{70} - 23730 q^{71} - 3264 q^{72} - 5671 q^{73} - 2728 q^{74} - 6462 q^{75} + 340 q^{76} + 702 q^{77} - 264 q^{78} + 6614 q^{79} + 1344 q^{80} - 630 q^{81} + 8232 q^{82} + 1002 q^{83} - 1488 q^{84} + 1008 q^{85} - 412 q^{86} + 6408 q^{87} + 336 q^{88} + 6258 q^{89} + 864 q^{90} - 5744 q^{91} + 288 q^{92} + 5268 q^{93} - 8376 q^{94} - 27672 q^{95} - 768 q^{96} - 26290 q^{97} - 10764 q^{98} - 14526 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(342))$$

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
342.4.a $$\chi_{342}(1, \cdot)$$ 342.4.a.a 1 1
342.4.a.b 1
342.4.a.c 1
342.4.a.d 1
342.4.a.e 1
342.4.a.f 2
342.4.a.g 2
342.4.a.h 2
342.4.a.i 2
342.4.a.j 2
342.4.a.k 2
342.4.a.l 3
342.4.a.m 3
342.4.b $$\chi_{342}(341, \cdot)$$ 342.4.b.a 10 1
342.4.b.b 10
342.4.e $$\chi_{342}(115, \cdot)$$ n/a 108 2
342.4.f $$\chi_{342}(7, \cdot)$$ n/a 120 2
342.4.g $$\chi_{342}(163, \cdot)$$ 342.4.g.a 2 2
342.4.g.b 2
342.4.g.c 2
342.4.g.d 2
342.4.g.e 4
342.4.g.f 6
342.4.g.g 6
342.4.g.h 6
342.4.g.i 10
342.4.g.j 10
342.4.h $$\chi_{342}(121, \cdot)$$ n/a 120 2
342.4.j $$\chi_{342}(65, \cdot)$$ n/a 120 2
342.4.n $$\chi_{342}(293, \cdot)$$ n/a 120 2
342.4.p $$\chi_{342}(113, \cdot)$$ n/a 120 2
342.4.s $$\chi_{342}(107, \cdot)$$ 342.4.s.a 20 2
342.4.s.b 20
342.4.u $$\chi_{342}(55, \cdot)$$ n/a 150 6
342.4.v $$\chi_{342}(25, \cdot)$$ n/a 360 6
342.4.w $$\chi_{342}(43, \cdot)$$ n/a 360 6
342.4.x $$\chi_{342}(29, \cdot)$$ n/a 360 6
342.4.bb $$\chi_{342}(53, \cdot)$$ n/a 120 6
342.4.bf $$\chi_{342}(155, \cdot)$$ n/a 360 6

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(342))$$ into lower level spaces

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