## 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

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

## 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

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