# Properties

 Label 336.3 Level 336 Weight 3 Dimension 2440 Nonzero newspaces 16 Newform subspaces 57 Sturm bound 18432 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$16$$ Newform subspaces: $$57$$ Sturm bound: $$18432$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(336))$$.

Total New Old
Modular forms 6480 2528 3952
Cusp forms 5808 2440 3368
Eisenstein series 672 88 584

## Trace form

 $$2440 q - 7 q^{3} - 40 q^{4} - 24 q^{5} - 20 q^{6} - 28 q^{7} + 24 q^{8} + 19 q^{9} + O(q^{10})$$ $$2440 q - 7 q^{3} - 40 q^{4} - 24 q^{5} - 20 q^{6} - 28 q^{7} + 24 q^{8} + 19 q^{9} + 136 q^{10} - 64 q^{11} + 100 q^{12} + 32 q^{13} + 44 q^{14} + 54 q^{15} - 56 q^{16} + 24 q^{17} + 84 q^{18} + 82 q^{19} - 160 q^{20} - 61 q^{21} - 192 q^{22} + 112 q^{23} + 28 q^{24} - 168 q^{25} + 200 q^{26} - 52 q^{27} - 24 q^{28} - 136 q^{29} - 324 q^{30} + 10 q^{31} - 320 q^{32} - 83 q^{33} - 488 q^{34} + 192 q^{35} - 440 q^{36} + 278 q^{37} - 336 q^{38} + 262 q^{39} - 296 q^{40} + 216 q^{41} + 200 q^{42} + 208 q^{43} + 880 q^{44} + 513 q^{45} + 1184 q^{46} + 288 q^{47} + 892 q^{48} - 24 q^{49} + 1312 q^{50} - 145 q^{51} + 1360 q^{52} + 584 q^{53} + 484 q^{54} + 396 q^{55} + 56 q^{56} - 98 q^{57} + 320 q^{58} + 64 q^{59} + 444 q^{60} + 6 q^{61} - 960 q^{62} - 339 q^{63} - 280 q^{64} - 560 q^{65} - 332 q^{66} - 1086 q^{67} - 664 q^{68} - 378 q^{69} - 1392 q^{70} - 1216 q^{71} - 276 q^{72} - 570 q^{73} - 696 q^{74} - 570 q^{75} - 1720 q^{76} - 512 q^{77} - 1288 q^{78} - 758 q^{79} - 1104 q^{80} + 99 q^{81} - 1416 q^{82} + 320 q^{83} - 964 q^{84} - 620 q^{85} - 1056 q^{86} - 60 q^{87} - 1688 q^{88} - 456 q^{89} - 1980 q^{90} + 1480 q^{91} - 1448 q^{92} - 187 q^{93} - 2424 q^{94} + 2880 q^{95} - 1828 q^{96} + 192 q^{97} - 1744 q^{98} + 646 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(336))$$

We only show spaces with odd 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
336.3.d $$\chi_{336}(113, \cdot)$$ 336.3.d.a 4 1
336.3.d.b 4
336.3.d.c 4
336.3.d.d 12
336.3.e $$\chi_{336}(167, \cdot)$$ None 0 1
336.3.f $$\chi_{336}(97, \cdot)$$ 336.3.f.a 2 1
336.3.f.b 2
336.3.f.c 4
336.3.f.d 8
336.3.g $$\chi_{336}(295, \cdot)$$ None 0 1
336.3.l $$\chi_{336}(265, \cdot)$$ None 0 1
336.3.m $$\chi_{336}(127, \cdot)$$ 336.3.m.a 4 1
336.3.m.b 4
336.3.m.c 4
336.3.n $$\chi_{336}(281, \cdot)$$ None 0 1
336.3.o $$\chi_{336}(335, \cdot)$$ 336.3.o.a 2 1
336.3.o.b 2
336.3.o.c 2
336.3.o.d 2
336.3.o.e 4
336.3.o.f 4
336.3.o.g 8
336.3.o.h 8
336.3.r $$\chi_{336}(13, \cdot)$$ 336.3.r.a 4 2
336.3.r.b 124
336.3.t $$\chi_{336}(29, \cdot)$$ 336.3.t.a 192 2
336.3.v $$\chi_{336}(83, \cdot)$$ 336.3.v.a 248 2
336.3.x $$\chi_{336}(43, \cdot)$$ 336.3.x.a 96 2
336.3.z $$\chi_{336}(47, \cdot)$$ 336.3.z.a 2 2
336.3.z.b 2
336.3.z.c 2
336.3.z.d 2
336.3.z.e 16
336.3.z.f 20
336.3.z.g 20
336.3.ba $$\chi_{336}(137, \cdot)$$ None 0 2
336.3.be $$\chi_{336}(79, \cdot)$$ 336.3.be.a 4 2
336.3.be.b 4
336.3.be.c 6
336.3.be.d 6
336.3.be.e 6
336.3.be.f 6
336.3.bf $$\chi_{336}(73, \cdot)$$ None 0 2
336.3.bg $$\chi_{336}(151, \cdot)$$ None 0 2
336.3.bh $$\chi_{336}(145, \cdot)$$ 336.3.bh.a 2 2
336.3.bh.b 2
336.3.bh.c 2
336.3.bh.d 2
336.3.bh.e 4
336.3.bh.f 4
336.3.bh.g 8
336.3.bh.h 8
336.3.bm $$\chi_{336}(215, \cdot)$$ None 0 2
336.3.bn $$\chi_{336}(65, \cdot)$$ 336.3.bn.a 2 2
336.3.bn.b 2
336.3.bn.c 4
336.3.bn.d 4
336.3.bn.e 4
336.3.bn.f 4
336.3.bn.g 8
336.3.bn.h 32
336.3.bp $$\chi_{336}(67, \cdot)$$ 336.3.bp.a 256 4
336.3.br $$\chi_{336}(59, \cdot)$$ 336.3.br.a 496 4
336.3.bt $$\chi_{336}(53, \cdot)$$ 336.3.bt.a 496 4
336.3.bv $$\chi_{336}(61, \cdot)$$ 336.3.bv.a 256 4

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(336))$$ into lower level spaces

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