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Properties

Label 392.8
Level 392
Weight 8
Dimension 17678
Nonzero newspaces 12
Sturm bound 75264
Trace bound 3

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Defining parameters

Level: \( N \) = \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(75264\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(392))\).

Total New Old
Modular forms 33288 17872 15416
Cusp forms 32568 17678 14890
Eisenstein series 720 194 526

Trace form

\( 17678 q - 24 q^{2} - 70 q^{3} + 86 q^{4} + 348 q^{5} + 238 q^{6} - 36 q^{7} + 1458 q^{8} - 2884 q^{9} + O(q^{10}) \) \( 17678 q - 24 q^{2} - 70 q^{3} + 86 q^{4} + 348 q^{5} + 238 q^{6} - 36 q^{7} + 1458 q^{8} - 2884 q^{9} - 1686 q^{10} + 3366 q^{11} - 4118 q^{12} - 31024 q^{13} - 36 q^{14} + 64962 q^{15} + 35314 q^{16} + 35976 q^{17} - 115336 q^{18} - 123014 q^{19} + 347838 q^{20} - 26544 q^{21} - 376814 q^{22} - 395874 q^{23} - 314326 q^{24} + 88196 q^{25} + 501942 q^{26} + 612182 q^{27} + 834540 q^{28} + 222636 q^{29} - 155566 q^{30} - 1207010 q^{31} - 1370394 q^{32} - 1784056 q^{33} - 1731946 q^{34} + 1236780 q^{35} + 2726730 q^{36} + 1379424 q^{37} + 1577346 q^{38} - 2478762 q^{39} + 16934 q^{40} - 2965020 q^{41} + 1679580 q^{42} - 2002414 q^{43} - 7293510 q^{44} + 6422320 q^{45} - 1863130 q^{46} + 2154270 q^{47} + 13837050 q^{48} - 1586418 q^{49} + 9038778 q^{50} - 10876466 q^{51} - 787762 q^{52} - 350856 q^{53} - 24638446 q^{54} + 20096038 q^{55} - 6597354 q^{56} - 1678020 q^{57} + 86950 q^{58} - 8974110 q^{59} + 31245894 q^{60} + 521036 q^{61} + 21580038 q^{62} - 8861064 q^{63} + 8683550 q^{64} + 6424812 q^{65} - 17634078 q^{66} + 8666758 q^{67} - 30335190 q^{68} + 10093880 q^{69} + 24762132 q^{70} - 13004862 q^{71} + 28171766 q^{72} - 5567568 q^{73} - 52732974 q^{74} - 23401310 q^{75} - 57637670 q^{76} - 4288806 q^{77} - 44738302 q^{78} + 28642558 q^{79} + 32559774 q^{80} + 20245752 q^{81} + 84362254 q^{82} + 86607006 q^{83} + 59484960 q^{84} + 19072252 q^{85} + 52847862 q^{86} - 71338842 q^{87} - 30247262 q^{88} - 56397996 q^{89} - 153991234 q^{90} - 65131524 q^{91} - 85996866 q^{92} - 36935672 q^{93} - 97935702 q^{94} + 18358758 q^{95} - 69972430 q^{96} + 172527432 q^{97} + 76259172 q^{98} + 182020048 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(392))\)

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
392.8.a \(\chi_{392}(1, \cdot)\) 392.8.a.a 1 1
392.8.a.b 1
392.8.a.c 1
392.8.a.d 1
392.8.a.e 2
392.8.a.f 3
392.8.a.g 3
392.8.a.h 4
392.8.a.i 6
392.8.a.j 7
392.8.a.k 7
392.8.a.l 7
392.8.a.m 7
392.8.a.n 10
392.8.a.o 12
392.8.b \(\chi_{392}(197, \cdot)\) n/a 282 1
392.8.e \(\chi_{392}(195, \cdot)\) n/a 276 1
392.8.f \(\chi_{392}(391, \cdot)\) None 0 1
392.8.i \(\chi_{392}(177, \cdot)\) n/a 140 2
392.8.l \(\chi_{392}(31, \cdot)\) None 0 2
392.8.m \(\chi_{392}(19, \cdot)\) n/a 552 2
392.8.p \(\chi_{392}(165, \cdot)\) n/a 552 2
392.8.q \(\chi_{392}(57, \cdot)\) n/a 588 6
392.8.t \(\chi_{392}(55, \cdot)\) None 0 6
392.8.u \(\chi_{392}(27, \cdot)\) n/a 2340 6
392.8.x \(\chi_{392}(29, \cdot)\) n/a 2340 6
392.8.y \(\chi_{392}(9, \cdot)\) n/a 1176 12
392.8.z \(\chi_{392}(37, \cdot)\) n/a 4680 12
392.8.bc \(\chi_{392}(3, \cdot)\) n/a 4680 12
392.8.bd \(\chi_{392}(47, \cdot)\) None 0 12

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(392))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(392)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 2}\)