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Properties

Label 336.8
Level 336
Weight 8
Dimension 8648
Nonzero newspaces 16
Sturm bound 49152
Trace bound 8

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Defining parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(49152\)
Trace bound: \(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(336))\).

Total New Old
Modular forms 21840 8740 13100
Cusp forms 21168 8648 12520
Eisenstein series 672 92 580

Trace form

\( 8648 q + 49 q^{3} - 744 q^{4} + 556 q^{5} - 356 q^{6} + 310 q^{7} - 4008 q^{8} - 3501 q^{9} + O(q^{10}) \) \( 8648 q + 49 q^{3} - 744 q^{4} + 556 q^{5} - 356 q^{6} + 310 q^{7} - 4008 q^{8} - 3501 q^{9} + 51880 q^{10} + 7224 q^{11} - 54716 q^{12} - 6060 q^{13} + 44028 q^{14} - 67518 q^{15} + 105544 q^{16} - 2908 q^{17} - 83180 q^{18} + 423530 q^{19} - 164000 q^{20} - 33271 q^{21} + 199520 q^{22} + 15860 q^{23} + 552732 q^{24} + 120668 q^{25} - 727960 q^{26} - 554828 q^{27} - 129944 q^{28} - 137596 q^{29} + 330748 q^{30} + 1987198 q^{31} + 1142400 q^{32} + 1269197 q^{33} + 2377944 q^{34} - 1059396 q^{35} - 1659448 q^{36} - 1512954 q^{37} - 3843152 q^{38} + 481352 q^{39} - 1681576 q^{40} - 67404 q^{41} - 239160 q^{42} + 743392 q^{43} + 9559472 q^{44} + 4850277 q^{45} + 9494848 q^{46} - 1780452 q^{47} - 14789316 q^{48} + 21617960 q^{49} - 22514976 q^{50} - 830547 q^{51} + 12424144 q^{52} + 14313068 q^{53} + 22994708 q^{54} + 4112736 q^{55} + 16622424 q^{56} + 2241354 q^{57} - 544320 q^{58} - 9033416 q^{59} - 14608324 q^{60} - 4590866 q^{61} - 27833664 q^{62} - 5144943 q^{63} - 27106776 q^{64} - 1923576 q^{65} + 23043220 q^{66} - 26846342 q^{67} + 41192168 q^{68} + 19810468 q^{69} - 8754512 q^{70} + 7818916 q^{71} - 23519060 q^{72} - 7158026 q^{73} - 37166936 q^{74} - 11996272 q^{75} + 31791624 q^{76} + 19878608 q^{77} + 63812472 q^{78} - 37450098 q^{79} + 52728176 q^{80} + 54298787 q^{81} + 27861304 q^{82} - 6919704 q^{83} + 5064860 q^{84} + 14768956 q^{85} - 64615840 q^{86} + 79456860 q^{87} - 192984728 q^{88} - 35590668 q^{89} - 56729212 q^{90} - 24905732 q^{91} + 144333272 q^{92} - 584819 q^{93} + 143913352 q^{94} + 123039292 q^{95} - 41289380 q^{96} + 92477132 q^{97} - 53327696 q^{98} + 58349102 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(336))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
336.8.a \(\chi_{336}(1, \cdot)\) 336.8.a.a 1 1
336.8.a.b 1
336.8.a.c 1
336.8.a.d 1
336.8.a.e 1
336.8.a.f 1
336.8.a.g 1
336.8.a.h 1
336.8.a.i 1
336.8.a.j 1
336.8.a.k 2
336.8.a.l 2
336.8.a.m 2
336.8.a.n 2
336.8.a.o 2
336.8.a.p 2
336.8.a.q 2
336.8.a.r 3
336.8.a.s 3
336.8.a.t 3
336.8.a.u 3
336.8.a.v 3
336.8.a.w 3
336.8.b \(\chi_{336}(223, \cdot)\) 336.8.b.a 10 1
336.8.b.b 10
336.8.b.c 18
336.8.b.d 18
336.8.c \(\chi_{336}(169, \cdot)\) None 0 1
336.8.h \(\chi_{336}(239, \cdot)\) 336.8.h.a 28 1
336.8.h.b 56
336.8.i \(\chi_{336}(41, \cdot)\) None 0 1
336.8.j \(\chi_{336}(71, \cdot)\) None 0 1
336.8.k \(\chi_{336}(209, \cdot)\) n/a 110 1
336.8.p \(\chi_{336}(55, \cdot)\) None 0 1
336.8.q \(\chi_{336}(193, \cdot)\) n/a 112 2
336.8.s \(\chi_{336}(155, \cdot)\) n/a 672 2
336.8.u \(\chi_{336}(139, \cdot)\) n/a 448 2
336.8.w \(\chi_{336}(85, \cdot)\) n/a 336 2
336.8.y \(\chi_{336}(125, \cdot)\) n/a 888 2
336.8.bb \(\chi_{336}(103, \cdot)\) None 0 2
336.8.bc \(\chi_{336}(17, \cdot)\) n/a 220 2
336.8.bd \(\chi_{336}(23, \cdot)\) None 0 2
336.8.bi \(\chi_{336}(89, \cdot)\) None 0 2
336.8.bj \(\chi_{336}(95, \cdot)\) n/a 224 2
336.8.bk \(\chi_{336}(25, \cdot)\) None 0 2
336.8.bl \(\chi_{336}(31, \cdot)\) n/a 112 2
336.8.bo \(\chi_{336}(5, \cdot)\) n/a 1776 4
336.8.bq \(\chi_{336}(37, \cdot)\) n/a 896 4
336.8.bs \(\chi_{336}(19, \cdot)\) n/a 896 4
336.8.bu \(\chi_{336}(11, \cdot)\) n/a 1776 4

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(336))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(336)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 20}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(336))\)\(^{\oplus 1}\)