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Properties

Label 392.6
Level 392
Weight 6
Dimension 12599
Nonzero newspaces 12
Sturm bound 56448
Trace bound 3

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Defining parameters

Level: \( N \) = \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(56448\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(392))\).

Total New Old
Modular forms 23880 12793 11087
Cusp forms 23160 12599 10561
Eisenstein series 720 194 526

Trace form

\( 12599 q - 32 q^{2} - 10 q^{3} - 10 q^{4} - 74 q^{5} - 146 q^{6} - 36 q^{7} - 302 q^{8} - 1207 q^{9} + O(q^{10}) \) \( 12599 q - 32 q^{2} - 10 q^{3} - 10 q^{4} - 74 q^{5} - 146 q^{6} - 36 q^{7} - 302 q^{8} - 1207 q^{9} + 602 q^{10} + 154 q^{11} + 1546 q^{12} + 2506 q^{13} - 36 q^{14} - 486 q^{15} - 3342 q^{16} - 4310 q^{17} + 1808 q^{18} - 11434 q^{19} + 9598 q^{20} + 6216 q^{21} + 26770 q^{22} + 13974 q^{23} - 8470 q^{24} - 8429 q^{25} - 30362 q^{26} - 36754 q^{27} - 33684 q^{28} - 3282 q^{29} - 18574 q^{30} + 14254 q^{31} + 50438 q^{32} + 14888 q^{33} + 98950 q^{34} + 2484 q^{35} + 43242 q^{36} + 3474 q^{37} - 66526 q^{38} + 15294 q^{39} - 21498 q^{40} + 67174 q^{41} + 46620 q^{42} - 73082 q^{43} - 102 q^{44} + 47290 q^{45} - 120890 q^{46} + 108126 q^{47} - 246534 q^{48} - 6294 q^{49} - 100094 q^{50} - 170234 q^{51} + 112558 q^{52} - 51542 q^{53} + 485714 q^{54} + 4302 q^{55} + 124662 q^{56} - 142428 q^{57} + 161206 q^{58} - 171290 q^{59} - 46554 q^{60} - 102146 q^{61} - 275386 q^{62} + 204888 q^{63} - 453154 q^{64} + 183944 q^{65} - 645438 q^{66} + 207026 q^{67} - 628086 q^{68} + 44648 q^{69} + 295956 q^{70} + 242858 q^{71} + 1142966 q^{72} + 213770 q^{73} + 1016194 q^{74} + 458230 q^{75} + 531194 q^{76} - 46386 q^{77} - 318430 q^{78} - 788594 q^{79} - 1020738 q^{80} - 1120899 q^{81} - 771874 q^{82} - 550726 q^{83} - 734976 q^{84} + 135608 q^{85} - 1132010 q^{86} + 1204686 q^{87} - 421982 q^{88} + 305782 q^{89} + 857774 q^{90} + 332868 q^{91} + 1652286 q^{92} - 526616 q^{93} + 2215914 q^{94} - 99746 q^{95} + 1119602 q^{96} + 490010 q^{97} - 93132 q^{98} - 2019596 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(392))\)

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
392.6.a \(\chi_{392}(1, \cdot)\) 392.6.a.a 1 1
392.6.a.b 1
392.6.a.c 1
392.6.a.d 2
392.6.a.e 2
392.6.a.f 2
392.6.a.g 4
392.6.a.h 4
392.6.a.i 5
392.6.a.j 5
392.6.a.k 5
392.6.a.l 5
392.6.a.m 6
392.6.a.n 8
392.6.b \(\chi_{392}(197, \cdot)\) n/a 200 1
392.6.e \(\chi_{392}(195, \cdot)\) n/a 196 1
392.6.f \(\chi_{392}(391, \cdot)\) None 0 1
392.6.i \(\chi_{392}(177, \cdot)\) 392.6.i.a 2 2
392.6.i.b 2
392.6.i.c 2
392.6.i.d 2
392.6.i.e 2
392.6.i.f 2
392.6.i.g 4
392.6.i.h 4
392.6.i.i 4
392.6.i.j 4
392.6.i.k 4
392.6.i.l 4
392.6.i.m 8
392.6.i.n 8
392.6.i.o 10
392.6.i.p 10
392.6.i.q 12
392.6.i.r 16
392.6.l \(\chi_{392}(31, \cdot)\) None 0 2
392.6.m \(\chi_{392}(19, \cdot)\) n/a 392 2
392.6.p \(\chi_{392}(165, \cdot)\) n/a 392 2
392.6.q \(\chi_{392}(57, \cdot)\) n/a 420 6
392.6.t \(\chi_{392}(55, \cdot)\) None 0 6
392.6.u \(\chi_{392}(27, \cdot)\) n/a 1668 6
392.6.x \(\chi_{392}(29, \cdot)\) n/a 1668 6
392.6.y \(\chi_{392}(9, \cdot)\) n/a 840 12
392.6.z \(\chi_{392}(37, \cdot)\) n/a 3336 12
392.6.bc \(\chi_{392}(3, \cdot)\) n/a 3336 12
392.6.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 \(S_{6}^{\mathrm{old}}(\Gamma_1(392))\) into lower level spaces

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