## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$24576$$ Trace bound: $$8$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(336))$$.

Total New Old
Modular forms 9552 3772 5780
Cusp forms 8880 3680 5200
Eisenstein series 672 92 580

## Trace form

 $$3680q + q^{3} + 24q^{4} - 4q^{5} - 68q^{6} - 42q^{7} - 168q^{8} - 117q^{9} + O(q^{10})$$ $$3680q + q^{3} + 24q^{4} - 4q^{5} - 68q^{6} - 42q^{7} - 168q^{8} - 117q^{9} - 280q^{10} + 120q^{11} + 196q^{12} + 4q^{13} + 348q^{14} - 318q^{15} + 584q^{16} + 52q^{17} - 44q^{18} + 458q^{19} - 160q^{20} + 425q^{21} - 1376q^{22} + 244q^{23} - 228q^{24} - 572q^{25} + 40q^{26} + 148q^{27} + 616q^{28} - 716q^{29} + 1468q^{30} + 702q^{31} + 1920q^{32} - 1075q^{33} + 344q^{34} + 924q^{35} + 968q^{36} - 746q^{37} - 2512q^{38} - 1384q^{39} - 4776q^{40} - 828q^{41} - 3960q^{42} - 2048q^{43} - 6992q^{44} + 117q^{45} - 2688q^{46} + 540q^{47} - 1476q^{48} + 8448q^{49} + 2784q^{50} + 2205q^{51} + 6608q^{52} + 2236q^{53} + 5492q^{54} + 96q^{55} + 7896q^{56} + 522q^{57} + 5184q^{58} + 3320q^{59} + 6716q^{60} + 3742q^{61} + 4800q^{62} + 177q^{63} + 1320q^{64} - 4056q^{65} - 2540q^{66} + 6554q^{67} - 8984q^{68} - 764q^{69} - 7952q^{70} + 356q^{71} + 2476q^{72} + 9638q^{73} + 5480q^{74} - 2752q^{75} + 4872q^{76} - 2672q^{77} + 6456q^{78} - 12082q^{79} - 1424q^{80} + 2939q^{81} - 1480q^{82} - 11736q^{83} + 284q^{84} - 10404q^{85} + 3424q^{86} - 1236q^{87} - 9880q^{88} + 132q^{89} - 6652q^{90} - 6148q^{91} - 4136q^{92} + 2941q^{93} + 6280q^{94} - 7108q^{95} - 6308q^{96} + 1276q^{97} + 5168q^{98} - 9682q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(336))$$

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
336.4.a $$\chi_{336}(1, \cdot)$$ 336.4.a.a 1 1
336.4.a.b 1
336.4.a.c 1
336.4.a.d 1
336.4.a.e 1
336.4.a.f 1
336.4.a.g 1
336.4.a.h 1
336.4.a.i 1
336.4.a.j 1
336.4.a.k 1
336.4.a.l 1
336.4.a.m 2
336.4.a.n 2
336.4.a.o 2
336.4.b $$\chi_{336}(223, \cdot)$$ 336.4.b.a 2 1
336.4.b.b 2
336.4.b.c 2
336.4.b.d 2
336.4.b.e 8
336.4.b.f 8
336.4.c $$\chi_{336}(169, \cdot)$$ None 0 1
336.4.h $$\chi_{336}(239, \cdot)$$ 336.4.h.a 12 1
336.4.h.b 24
336.4.i $$\chi_{336}(41, \cdot)$$ None 0 1
336.4.j $$\chi_{336}(71, \cdot)$$ None 0 1
336.4.k $$\chi_{336}(209, \cdot)$$ 336.4.k.a 2 1
336.4.k.b 4
336.4.k.c 8
336.4.k.d 8
336.4.k.e 24
336.4.p $$\chi_{336}(55, \cdot)$$ None 0 1
336.4.q $$\chi_{336}(193, \cdot)$$ 336.4.q.a 2 2
336.4.q.b 2
336.4.q.c 2
336.4.q.d 2
336.4.q.e 2
336.4.q.f 2
336.4.q.g 4
336.4.q.h 4
336.4.q.i 4
336.4.q.j 4
336.4.q.k 6
336.4.q.l 6
336.4.q.m 8
336.4.s $$\chi_{336}(155, \cdot)$$ n/a 288 2
336.4.u $$\chi_{336}(139, \cdot)$$ n/a 192 2
336.4.w $$\chi_{336}(85, \cdot)$$ n/a 144 2
336.4.y $$\chi_{336}(125, \cdot)$$ n/a 376 2
336.4.bb $$\chi_{336}(103, \cdot)$$ None 0 2
336.4.bc $$\chi_{336}(17, \cdot)$$ 336.4.bc.a 2 2
336.4.bc.b 2
336.4.bc.c 12
336.4.bc.d 12
336.4.bc.e 16
336.4.bc.f 48
336.4.bd $$\chi_{336}(23, \cdot)$$ None 0 2
336.4.bi $$\chi_{336}(89, \cdot)$$ None 0 2
336.4.bj $$\chi_{336}(95, \cdot)$$ 336.4.bj.a 2 2
336.4.bj.b 2
336.4.bj.c 2
336.4.bj.d 2
336.4.bj.e 28
336.4.bj.f 28
336.4.bj.g 32
336.4.bk $$\chi_{336}(25, \cdot)$$ None 0 2
336.4.bl $$\chi_{336}(31, \cdot)$$ 336.4.bl.a 2 2
336.4.bl.b 2
336.4.bl.c 2
336.4.bl.d 2
336.4.bl.e 6
336.4.bl.f 6
336.4.bl.g 6
336.4.bl.h 6
336.4.bl.i 8
336.4.bl.j 8
336.4.bo $$\chi_{336}(5, \cdot)$$ n/a 752 4
336.4.bq $$\chi_{336}(37, \cdot)$$ n/a 384 4
336.4.bs $$\chi_{336}(19, \cdot)$$ n/a 384 4
336.4.bu $$\chi_{336}(11, \cdot)$$ n/a 752 4

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(336))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(336)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$