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Properties

Label 336.9
Level 336
Weight 9
Dimension 9892
Nonzero newspaces 16
Sturm bound 55296
Trace bound 9

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Defining parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(55296\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(336))\).

Total New Old
Modular forms 24912 9980 14932
Cusp forms 24240 9892 14348
Eisenstein series 672 88 584

Trace form

\( 9892 q - 7 q^{3} + 728 q^{4} - 2016 q^{5} + 6460 q^{6} + 1484 q^{7} - 34920 q^{8} + 31183 q^{9} + O(q^{10}) \) \( 9892 q - 7 q^{3} + 728 q^{4} - 2016 q^{5} + 6460 q^{6} + 1484 q^{7} - 34920 q^{8} + 31183 q^{9} - 7448 q^{10} - 79104 q^{11} + 28132 q^{12} - 119992 q^{13} - 116772 q^{14} + 183798 q^{15} + 556744 q^{16} - 231840 q^{17} + 633204 q^{18} + 914290 q^{19} - 1380000 q^{20} - 421069 q^{21} + 2124384 q^{22} - 4064832 q^{23} + 1578268 q^{24} + 1955820 q^{25} + 3364200 q^{26} + 1329212 q^{27} - 2069784 q^{28} + 652896 q^{29} - 3478500 q^{30} - 468422 q^{31} - 2815680 q^{32} + 2203117 q^{33} + 1903576 q^{34} + 6587136 q^{35} - 544952 q^{36} + 9338126 q^{37} - 4826640 q^{38} + 5302198 q^{39} - 18067496 q^{40} - 26944416 q^{41} + 15869480 q^{42} + 11383456 q^{43} - 89017872 q^{44} - 10609767 q^{45} + 78467264 q^{46} + 30395520 q^{47} + 60256252 q^{48} - 258653916 q^{49} - 73520352 q^{50} - 32513449 q^{51} - 226378160 q^{52} + 1380768 q^{53} - 14671532 q^{54} + 96193548 q^{55} + 122438904 q^{56} + 101625214 q^{57} + 54048512 q^{58} - 22469376 q^{59} - 73264452 q^{60} - 133690674 q^{61} - 199894464 q^{62} - 102399531 q^{63} - 66504088 q^{64} - 145665216 q^{65} - 170385164 q^{66} + 210241074 q^{67} + 445254120 q^{68} + 61125942 q^{69} + 266296752 q^{70} - 383109888 q^{71} + 37100652 q^{72} + 258952734 q^{73} - 281308056 q^{74} + 321544398 q^{75} - 106849720 q^{76} + 55698432 q^{77} - 345477160 q^{78} + 142742074 q^{79} - 76153296 q^{80} + 316755135 q^{81} + 324989688 q^{82} - 418656000 q^{83} + 120237308 q^{84} - 256644092 q^{85} + 246903648 q^{86} + 319100340 q^{87} - 315230360 q^{88} + 37935072 q^{89} - 457433724 q^{90} + 489982216 q^{91} - 1865467944 q^{92} - 890600803 q^{93} + 1200589320 q^{94} - 2602955520 q^{95} + 1318052828 q^{96} + 240854904 q^{97} - 407778960 q^{98} + 536326966 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(336))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
336.9.d \(\chi_{336}(113, \cdot)\) 336.9.d.a 16 1
336.9.d.b 16
336.9.d.c 16
336.9.d.d 48
336.9.e \(\chi_{336}(167, \cdot)\) None 0 1
336.9.f \(\chi_{336}(97, \cdot)\) 336.9.f.a 10 1
336.9.f.b 10
336.9.f.c 12
336.9.f.d 32
336.9.g \(\chi_{336}(295, \cdot)\) None 0 1
336.9.l \(\chi_{336}(265, \cdot)\) None 0 1
336.9.m \(\chi_{336}(127, \cdot)\) 336.9.m.a 16 1
336.9.m.b 16
336.9.m.c 16
336.9.n \(\chi_{336}(281, \cdot)\) None 0 1
336.9.o \(\chi_{336}(335, \cdot)\) n/a 128 1
336.9.r \(\chi_{336}(13, \cdot)\) n/a 512 2
336.9.t \(\chi_{336}(29, \cdot)\) n/a 768 2
336.9.v \(\chi_{336}(83, \cdot)\) n/a 1016 2
336.9.x \(\chi_{336}(43, \cdot)\) n/a 384 2
336.9.z \(\chi_{336}(47, \cdot)\) n/a 256 2
336.9.ba \(\chi_{336}(137, \cdot)\) None 0 2
336.9.be \(\chi_{336}(79, \cdot)\) n/a 128 2
336.9.bf \(\chi_{336}(73, \cdot)\) None 0 2
336.9.bg \(\chi_{336}(151, \cdot)\) None 0 2
336.9.bh \(\chi_{336}(145, \cdot)\) n/a 128 2
336.9.bm \(\chi_{336}(215, \cdot)\) None 0 2
336.9.bn \(\chi_{336}(65, \cdot)\) n/a 252 2
336.9.bp \(\chi_{336}(67, \cdot)\) n/a 1024 4
336.9.br \(\chi_{336}(59, \cdot)\) n/a 2032 4
336.9.bt \(\chi_{336}(53, \cdot)\) n/a 2032 4
336.9.bv \(\chi_{336}(61, \cdot)\) n/a 1024 4

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

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(336))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(336)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 10}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 2}\)