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Properties

Label 396.6
Level 396
Weight 6
Dimension 9729
Nonzero newspaces 16
Sturm bound 51840
Trace bound 4

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Defining parameters

Level: \( N \) = \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(51840\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(396))\).

Total New Old
Modular forms 22000 9893 12107
Cusp forms 21200 9729 11471
Eisenstein series 800 164 636

Trace form

\( 9729 q - 9 q^{2} - 24 q^{3} + 27 q^{4} + 132 q^{5} + 34 q^{6} - 59 q^{7} - 15 q^{8} - 100 q^{9} + O(q^{10}) \) \( 9729 q - 9 q^{2} - 24 q^{3} + 27 q^{4} + 132 q^{5} + 34 q^{6} - 59 q^{7} - 15 q^{8} - 100 q^{9} - 1240 q^{10} - 82 q^{11} + 932 q^{12} - 263 q^{13} + 5716 q^{14} - 234 q^{15} - 2057 q^{16} - 7067 q^{17} - 4004 q^{18} + 5091 q^{19} + 10994 q^{20} - 286 q^{21} + 2508 q^{22} + 16199 q^{23} - 4490 q^{24} - 27608 q^{25} - 16240 q^{26} - 5874 q^{27} + 27248 q^{28} + 34685 q^{29} + 13744 q^{30} + 49298 q^{31} + 14436 q^{32} + 22867 q^{33} - 97124 q^{34} - 70711 q^{35} - 12818 q^{36} - 75680 q^{37} + 49264 q^{38} - 65388 q^{39} + 135106 q^{40} + 117069 q^{41} - 37468 q^{42} + 64076 q^{43} - 63190 q^{44} + 205892 q^{45} - 148510 q^{46} + 84293 q^{47} + 11842 q^{48} - 236579 q^{49} - 38437 q^{50} - 291514 q^{51} - 98272 q^{52} - 397075 q^{53} + 88406 q^{54} + 107352 q^{55} + 478148 q^{56} + 415584 q^{57} + 99634 q^{58} + 337998 q^{59} - 83380 q^{60} + 200649 q^{61} - 506490 q^{62} + 138190 q^{63} - 329973 q^{64} - 185460 q^{65} - 492852 q^{66} - 378391 q^{67} - 622296 q^{68} - 561582 q^{69} - 97766 q^{70} - 399580 q^{71} + 711712 q^{72} - 356561 q^{73} + 1376272 q^{74} + 391970 q^{75} + 993588 q^{76} + 916360 q^{77} + 683428 q^{78} + 339831 q^{79} - 200270 q^{80} - 351876 q^{81} - 681835 q^{82} + 102159 q^{83} - 451350 q^{84} + 689355 q^{85} - 843569 q^{86} + 142454 q^{87} - 825010 q^{88} - 583621 q^{89} - 470576 q^{90} - 847487 q^{91} - 669018 q^{92} - 873698 q^{93} + 462734 q^{94} - 255789 q^{95} + 77362 q^{96} - 624876 q^{97} - 823561 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(396))\)

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
396.6.a \(\chi_{396}(1, \cdot)\) 396.6.a.a 1 1
396.6.a.b 1
396.6.a.c 1
396.6.a.d 1
396.6.a.e 1
396.6.a.f 2
396.6.a.g 3
396.6.a.h 3
396.6.a.i 4
396.6.a.j 4
396.6.b \(\chi_{396}(197, \cdot)\) 396.6.b.a 20 1
396.6.c \(\chi_{396}(287, \cdot)\) 396.6.c.a 50 1
396.6.c.b 50
396.6.h \(\chi_{396}(307, \cdot)\) n/a 148 1
396.6.i \(\chi_{396}(133, \cdot)\) 396.6.i.a 48 2
396.6.i.b 52
396.6.j \(\chi_{396}(37, \cdot)\) 396.6.j.a 4 4
396.6.j.b 16
396.6.j.c 20
396.6.j.d 20
396.6.j.e 40
396.6.k \(\chi_{396}(43, \cdot)\) n/a 712 2
396.6.p \(\chi_{396}(23, \cdot)\) n/a 600 2
396.6.q \(\chi_{396}(65, \cdot)\) n/a 120 2
396.6.r \(\chi_{396}(19, \cdot)\) n/a 592 4
396.6.w \(\chi_{396}(71, \cdot)\) n/a 480 4
396.6.x \(\chi_{396}(17, \cdot)\) 396.6.x.a 80 4
396.6.y \(\chi_{396}(25, \cdot)\) n/a 480 8
396.6.z \(\chi_{396}(29, \cdot)\) n/a 480 8
396.6.ba \(\chi_{396}(47, \cdot)\) n/a 2848 8
396.6.bf \(\chi_{396}(7, \cdot)\) n/a 2848 8

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

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(396))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(396)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 18}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(44))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(99))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(132))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(396))\)\(^{\oplus 1}\)