# Properties

 Label 336.7 Level 336 Weight 7 Dimension 7408 Nonzero newspaces 16 Sturm bound 43008 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$16$$ Sturm bound: $$43008$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(336))$$.

Total New Old
Modular forms 18768 7496 11272
Cusp forms 18096 7408 10688
Eisenstein series 672 88 584

## Trace form

 $$7408 q - 7 q^{3} + 344 q^{4} - 264 q^{5} - 1028 q^{6} + 340 q^{7} + 3864 q^{8} + 3835 q^{9} + O(q^{10})$$ $$7408 q - 7 q^{3} + 344 q^{4} - 264 q^{5} - 1028 q^{6} + 340 q^{7} + 3864 q^{8} + 3835 q^{9} - 4504 q^{10} + 5440 q^{11} - 4700 q^{12} - 10032 q^{13} + 15388 q^{14} + 54 q^{15} - 28216 q^{16} + 14664 q^{17} - 29772 q^{18} - 494 q^{19} + 28000 q^{20} + 26723 q^{21} + 166496 q^{22} - 105904 q^{23} - 169700 q^{24} - 58288 q^{25} + 10600 q^{26} + 94700 q^{27} + 69096 q^{28} - 143960 q^{29} + 321756 q^{30} + 153642 q^{31} + 105920 q^{32} - 26195 q^{33} - 501288 q^{34} - 504768 q^{35} - 409016 q^{36} - 294234 q^{37} - 33936 q^{38} + 300646 q^{39} + 946904 q^{40} + 259848 q^{41} + 569960 q^{42} - 183248 q^{43} - 508048 q^{44} - 674607 q^{45} - 1555520 q^{46} - 376992 q^{47} - 302468 q^{48} - 3892592 q^{49} + 1549472 q^{50} - 515425 q^{51} + 2088272 q^{52} + 730648 q^{53} + 972244 q^{54} - 844404 q^{55} - 1122632 q^{56} - 1478306 q^{57} - 1119616 q^{58} + 443072 q^{59} - 3785796 q^{60} + 641142 q^{61} - 1253568 q^{62} + 17181 q^{63} + 3740648 q^{64} + 2599280 q^{65} + 3060532 q^{66} + 2026210 q^{67} - 880664 q^{68} - 346842 q^{69} - 4337232 q^{70} - 2595392 q^{71} - 1504020 q^{72} + 67798 q^{73} + 9789672 q^{74} - 747210 q^{75} + 7045064 q^{76} + 860672 q^{77} - 2281960 q^{78} - 3515094 q^{79} - 9336144 q^{80} + 7169739 q^{81} - 5329544 q^{82} + 4995520 q^{83} - 581572 q^{84} - 2522700 q^{85} - 8285856 q^{86} - 75996 q^{87} + 3069800 q^{88} + 7030824 q^{89} + 1598340 q^{90} + 2383688 q^{91} + 8515544 q^{92} + 6718613 q^{93} + 9584904 q^{94} - 13446720 q^{95} + 6957788 q^{96} - 1291120 q^{97} + 6674416 q^{98} + 1425958 q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(336))$$

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
336.7.d $$\chi_{336}(113, \cdot)$$ 336.7.d.a 12 1
336.7.d.b 12
336.7.d.c 12
336.7.d.d 36
336.7.e $$\chi_{336}(167, \cdot)$$ None 0 1
336.7.f $$\chi_{336}(97, \cdot)$$ 336.7.f.a 8 1
336.7.f.b 8
336.7.f.c 8
336.7.f.d 24
336.7.g $$\chi_{336}(295, \cdot)$$ None 0 1
336.7.l $$\chi_{336}(265, \cdot)$$ None 0 1
336.7.m $$\chi_{336}(127, \cdot)$$ 336.7.m.a 12 1
336.7.m.b 12
336.7.m.c 12
336.7.n $$\chi_{336}(281, \cdot)$$ None 0 1
336.7.o $$\chi_{336}(335, \cdot)$$ 336.7.o.a 2 1
336.7.o.b 2
336.7.o.c 2
336.7.o.d 2
336.7.o.e 24
336.7.o.f 64
336.7.r $$\chi_{336}(13, \cdot)$$ n/a 384 2
336.7.t $$\chi_{336}(29, \cdot)$$ n/a 576 2
336.7.v $$\chi_{336}(83, \cdot)$$ n/a 760 2
336.7.x $$\chi_{336}(43, \cdot)$$ n/a 288 2
336.7.z $$\chi_{336}(47, \cdot)$$ n/a 192 2
336.7.ba $$\chi_{336}(137, \cdot)$$ None 0 2
336.7.be $$\chi_{336}(79, \cdot)$$ 336.7.be.a 16 2
336.7.be.b 16
336.7.be.c 16
336.7.be.d 16
336.7.be.e 16
336.7.be.f 16
336.7.bf $$\chi_{336}(73, \cdot)$$ None 0 2
336.7.bg $$\chi_{336}(151, \cdot)$$ None 0 2
336.7.bh $$\chi_{336}(145, \cdot)$$ 336.7.bh.a 8 2
336.7.bh.b 8
336.7.bh.c 8
336.7.bh.d 8
336.7.bh.e 8
336.7.bh.f 8
336.7.bh.g 24
336.7.bh.h 24
336.7.bm $$\chi_{336}(215, \cdot)$$ None 0 2
336.7.bn $$\chi_{336}(65, \cdot)$$ n/a 188 2
336.7.bp $$\chi_{336}(67, \cdot)$$ n/a 768 4
336.7.br $$\chi_{336}(59, \cdot)$$ n/a 1520 4
336.7.bt $$\chi_{336}(53, \cdot)$$ n/a 1520 4
336.7.bv $$\chi_{336}(61, \cdot)$$ n/a 768 4

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

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(336))$$ into lower level spaces

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