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Properties

Label 392.4
Level 392
Weight 4
Dimension 7521
Nonzero newspaces 12
Sturm bound 37632
Trace bound 3

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Defining parameters

Level: \( N \) = \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(37632\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(392))\).

Total New Old
Modular forms 14472 7715 6757
Cusp forms 13752 7521 6231
Eisenstein series 720 194 526

Trace form

\( 7521 q - 32 q^{2} - 34 q^{3} - 42 q^{4} - 2 q^{5} - 2 q^{6} - 36 q^{7} - 14 q^{8} + 11 q^{9} + O(q^{10}) \) \( 7521 q - 32 q^{2} - 34 q^{3} - 42 q^{4} - 2 q^{5} - 2 q^{6} - 36 q^{7} - 14 q^{8} + 11 q^{9} - 86 q^{10} - 158 q^{11} - 86 q^{12} - 134 q^{13} - 36 q^{14} - 198 q^{15} - 14 q^{16} - 50 q^{17} - 352 q^{18} - 322 q^{19} - 962 q^{20} + 168 q^{21} - 318 q^{22} - 138 q^{23} + 938 q^{24} + 601 q^{25} + 1750 q^{26} + 1406 q^{27} + 1260 q^{28} + 198 q^{29} + 1874 q^{30} + 1806 q^{31} + 38 q^{32} + 920 q^{33} - 1082 q^{34} - 540 q^{35} - 2742 q^{36} - 1422 q^{37} - 3214 q^{38} - 3522 q^{39} - 1306 q^{40} - 2366 q^{41} + 1260 q^{42} - 3362 q^{43} + 3834 q^{44} - 3350 q^{45} + 2374 q^{46} - 450 q^{47} + 762 q^{48} + 342 q^{49} - 2222 q^{50} + 5734 q^{51} - 5234 q^{52} + 3706 q^{53} - 8302 q^{54} + 8398 q^{55} - 2346 q^{56} + 6012 q^{57} - 5082 q^{58} + 4414 q^{59} - 6906 q^{60} + 1558 q^{61} - 1882 q^{62} - 744 q^{63} + 1950 q^{64} - 2008 q^{65} + 3810 q^{66} - 10678 q^{67} - 3222 q^{68} - 12232 q^{69} - 1068 q^{70} - 10774 q^{71} - 394 q^{72} - 3554 q^{73} + 82 q^{74} - 1730 q^{75} + 4250 q^{76} + 1242 q^{77} + 9410 q^{78} + 4686 q^{79} + 14814 q^{80} - 7989 q^{81} + 11774 q^{82} + 5090 q^{83} + 8928 q^{84} + 7712 q^{85} + 14278 q^{86} + 9486 q^{87} + 9378 q^{88} + 4594 q^{89} + 11966 q^{90} + 7116 q^{91} + 6846 q^{92} + 11224 q^{93} + 9450 q^{94} + 16318 q^{95} + 2738 q^{96} - 4610 q^{97} - 5436 q^{98} + 1324 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(392))\)

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
392.4.a \(\chi_{392}(1, \cdot)\) 392.4.a.a 1 1
392.4.a.b 1
392.4.a.c 1
392.4.a.d 1
392.4.a.e 1
392.4.a.f 2
392.4.a.g 2
392.4.a.h 2
392.4.a.i 3
392.4.a.j 3
392.4.a.k 3
392.4.a.l 3
392.4.a.m 4
392.4.a.n 4
392.4.b \(\chi_{392}(197, \cdot)\) n/a 118 1
392.4.e \(\chi_{392}(195, \cdot)\) n/a 116 1
392.4.f \(\chi_{392}(391, \cdot)\) None 0 1
392.4.i \(\chi_{392}(177, \cdot)\) 392.4.i.a 2 2
392.4.i.b 2
392.4.i.c 2
392.4.i.d 2
392.4.i.e 2
392.4.i.f 2
392.4.i.g 2
392.4.i.h 2
392.4.i.i 4
392.4.i.j 4
392.4.i.k 4
392.4.i.l 4
392.4.i.m 6
392.4.i.n 6
392.4.i.o 8
392.4.i.p 8
392.4.l \(\chi_{392}(31, \cdot)\) None 0 2
392.4.m \(\chi_{392}(19, \cdot)\) n/a 232 2
392.4.p \(\chi_{392}(165, \cdot)\) n/a 232 2
392.4.q \(\chi_{392}(57, \cdot)\) n/a 252 6
392.4.t \(\chi_{392}(55, \cdot)\) None 0 6
392.4.u \(\chi_{392}(27, \cdot)\) n/a 996 6
392.4.x \(\chi_{392}(29, \cdot)\) n/a 996 6
392.4.y \(\chi_{392}(9, \cdot)\) n/a 504 12
392.4.z \(\chi_{392}(37, \cdot)\) n/a 1992 12
392.4.bc \(\chi_{392}(3, \cdot)\) n/a 1992 12
392.4.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 \(S_{4}^{\mathrm{old}}(\Gamma_1(392))\) into lower level spaces

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