## Defining parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Sturm bound: $$27840$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(354))$$.

Total New Old
Modular forms 10672 2608 8064
Cusp forms 10208 2608 7600
Eisenstein series 464 0 464

## Trace form

 $$2608q + 4q^{2} + 6q^{3} - 8q^{4} - 12q^{5} - 12q^{6} + 32q^{7} + 16q^{8} - 18q^{9} + O(q^{10})$$ $$2608q + 4q^{2} + 6q^{3} - 8q^{4} - 12q^{5} - 12q^{6} + 32q^{7} + 16q^{8} - 18q^{9} + 24q^{10} - 24q^{11} + 24q^{12} - 76q^{13} - 64q^{14} + 36q^{15} - 32q^{16} + 252q^{17} + 36q^{18} - 40q^{19} - 48q^{20} - 96q^{21} + 48q^{22} - 336q^{23} - 48q^{24} + 178q^{25} + 152q^{26} + 54q^{27} + 128q^{28} - 60q^{29} - 72q^{30} + 176q^{31} + 64q^{32} + 72q^{33} - 504q^{34} + 192q^{35} - 72q^{36} - 508q^{37} + 80q^{38} + 228q^{39} + 96q^{40} - 84q^{41} + 192q^{42} + 104q^{43} - 96q^{44} - 7300q^{45} - 10696q^{46} - 8392q^{47} + 96q^{48} - 7250q^{49} - 3604q^{50} - 292q^{51} + 1552q^{52} + 6100q^{53} + 7606q^{54} + 15864q^{55} - 256q^{56} + 16360q^{57} + 6848q^{58} + 14116q^{59} + 9888q^{60} + 13140q^{61} + 4984q^{62} + 11308q^{63} - 128q^{64} + 5112q^{65} + 262q^{66} - 5016q^{67} - 2704q^{68} - 11056q^{69} - 17552q^{70} - 24320q^{71} + 144q^{72} - 18068q^{73} - 20096q^{74} - 12946q^{75} - 160q^{76} + 384q^{77} - 456q^{78} + 1040q^{79} - 192q^{80} - 162q^{81} + 168q^{82} + 984q^{83} - 384q^{84} + 1512q^{85} - 208q^{86} + 180q^{87} + 192q^{88} - 1620q^{89} + 216q^{90} + 1216q^{91} - 1344q^{92} - 528q^{93} - 384q^{94} - 240q^{95} - 192q^{96} - 2308q^{97} - 348q^{98} - 216q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(354))$$

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
354.4.a $$\chi_{354}(1, \cdot)$$ 354.4.a.a 2 1
354.4.a.b 2
354.4.a.c 2
354.4.a.d 3
354.4.a.e 3
354.4.a.f 3
354.4.a.g 3
354.4.a.h 4
354.4.a.i 6
354.4.c $$\chi_{354}(353, \cdot)$$ 354.4.c.a 30 1
354.4.c.b 30
354.4.e $$\chi_{354}(7, \cdot)$$ n/a 840 28
354.4.g $$\chi_{354}(11, \cdot)$$ n/a 1680 28

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(354))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(354)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(59))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(118))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(177))$$$$^{\oplus 2}$$