## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newforms: $$56$$ Sturm bound: $$12288$$ Trace bound: $$8$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(336))$$.

Total New Old
Modular forms 3408 1288 2120
Cusp forms 2737 1196 1541
Eisenstein series 671 92 579

## Trace form

 $$1196q - 7q^{3} - 8q^{4} + 4q^{5} + 4q^{6} - 10q^{7} + 24q^{8} + 7q^{9} + O(q^{10})$$ $$1196q - 7q^{3} - 8q^{4} + 4q^{5} + 4q^{6} - 10q^{7} + 24q^{8} + 7q^{9} - 8q^{10} + 24q^{11} - 12q^{12} - 4q^{13} - 12q^{14} + 2q^{15} - 56q^{16} - 4q^{17} - 28q^{18} + 26q^{19} - 32q^{20} - 7q^{21} - 80q^{22} + 20q^{23} - 68q^{24} + 24q^{25} - 40q^{26} - 28q^{27} - 24q^{28} + 44q^{29} - 52q^{30} - 2q^{31} + 13q^{33} - 8q^{34} + 12q^{35} - 56q^{36} + 62q^{37} + 16q^{38} - 48q^{39} + 24q^{40} + 12q^{41} - 40q^{42} - 48q^{43} - 16q^{44} - 35q^{45} - 48q^{46} - 36q^{47} - 4q^{48} - 84q^{49} - 96q^{50} - 107q^{51} - 176q^{52} - 124q^{53} - 84q^{54} - 160q^{55} - 168q^{56} - 86q^{57} - 224q^{58} - 104q^{59} - 196q^{60} - 234q^{61} - 192q^{62} - 47q^{63} - 344q^{64} - 72q^{65} - 236q^{66} - 86q^{67} - 184q^{68} - 140q^{69} - 192q^{70} + 4q^{71} - 180q^{72} - 66q^{73} - 104q^{74} - 40q^{75} - 152q^{76} - 16q^{77} - 120q^{78} + 14q^{79} - 16q^{80} - 89q^{81} - 72q^{82} + 72q^{83} - 4q^{84} - 84q^{85} + 32q^{86} + 36q^{87} + 40q^{88} + 12q^{89} + 36q^{90} - 164q^{91} + 56q^{92} - 43q^{93} + 200q^{94} - 68q^{95} + 220q^{96} + 20q^{97} + 208q^{98} + 14q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(336))$$

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
336.2.a $$\chi_{336}(1, \cdot)$$ 336.2.a.a 1 1
336.2.a.b 1
336.2.a.c 1
336.2.a.d 1
336.2.a.e 1
336.2.a.f 1
336.2.b $$\chi_{336}(223, \cdot)$$ 336.2.b.a 2 1
336.2.b.b 2
336.2.b.c 2
336.2.b.d 2
336.2.c $$\chi_{336}(169, \cdot)$$ None 0 1
336.2.h $$\chi_{336}(239, \cdot)$$ 336.2.h.a 4 1
336.2.h.b 8
336.2.i $$\chi_{336}(41, \cdot)$$ None 0 1
336.2.j $$\chi_{336}(71, \cdot)$$ None 0 1
336.2.k $$\chi_{336}(209, \cdot)$$ 336.2.k.a 2 1
336.2.k.b 4
336.2.k.c 8
336.2.p $$\chi_{336}(55, \cdot)$$ None 0 1
336.2.q $$\chi_{336}(193, \cdot)$$ 336.2.q.a 2 2
336.2.q.b 2
336.2.q.c 2
336.2.q.d 2
336.2.q.e 2
336.2.q.f 2
336.2.q.g 4
336.2.s $$\chi_{336}(155, \cdot)$$ 336.2.s.a 4 2
336.2.s.b 4
336.2.s.c 40
336.2.s.d 48
336.2.u $$\chi_{336}(139, \cdot)$$ 336.2.u.a 64 2
336.2.w $$\chi_{336}(85, \cdot)$$ 336.2.w.a 20 2
336.2.w.b 28
336.2.y $$\chi_{336}(125, \cdot)$$ 336.2.y.a 120 2
336.2.bb $$\chi_{336}(103, \cdot)$$ None 0 2
336.2.bc $$\chi_{336}(17, \cdot)$$ 336.2.bc.a 2 2
336.2.bc.b 2
336.2.bc.c 2
336.2.bc.d 2
336.2.bc.e 4
336.2.bc.f 16
336.2.bd $$\chi_{336}(23, \cdot)$$ None 0 2
336.2.bi $$\chi_{336}(89, \cdot)$$ None 0 2
336.2.bj $$\chi_{336}(95, \cdot)$$ 336.2.bj.a 2 2
336.2.bj.b 2
336.2.bj.c 2
336.2.bj.d 2
336.2.bj.e 8
336.2.bj.f 8
336.2.bj.g 8
336.2.bk $$\chi_{336}(25, \cdot)$$ None 0 2
336.2.bl $$\chi_{336}(31, \cdot)$$ 336.2.bl.a 2 2
336.2.bl.b 2
336.2.bl.c 2
336.2.bl.d 2
336.2.bl.e 2
336.2.bl.f 2
336.2.bl.g 2
336.2.bl.h 2
336.2.bo $$\chi_{336}(5, \cdot)$$ 336.2.bo.a 240 4
336.2.bq $$\chi_{336}(37, \cdot)$$ 336.2.bq.a 8 4
336.2.bq.b 120
336.2.bs $$\chi_{336}(19, \cdot)$$ 336.2.bs.a 128 4
336.2.bu $$\chi_{336}(11, \cdot)$$ 336.2.bu.a 240 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(336))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(336)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$