Properties

Label 392.2
Level 392
Weight 2
Dimension 2442
Nonzero newspaces 12
Newform subspaces 52
Sturm bound 18816
Trace bound 3

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Defining parameters

Level: \( N \) = \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 52 \)
Sturm bound: \(18816\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(392))\).

Total New Old
Modular forms 5064 2636 2428
Cusp forms 4345 2442 1903
Eisenstein series 719 194 525

Trace form

\( 2442 q - 30 q^{2} - 30 q^{3} - 30 q^{4} - 30 q^{6} - 36 q^{7} - 54 q^{8} - 48 q^{9} + O(q^{10}) \) \( 2442 q - 30 q^{2} - 30 q^{3} - 30 q^{4} - 30 q^{6} - 36 q^{7} - 54 q^{8} - 48 q^{9} - 30 q^{10} - 18 q^{11} - 30 q^{12} + 12 q^{13} - 36 q^{14} - 30 q^{15} - 30 q^{16} - 48 q^{17} - 66 q^{18} - 30 q^{19} - 66 q^{20} - 114 q^{22} - 66 q^{23} - 102 q^{24} - 96 q^{25} - 90 q^{26} - 90 q^{27} - 84 q^{28} - 126 q^{30} - 66 q^{31} - 90 q^{32} - 72 q^{33} - 102 q^{34} - 36 q^{35} - 114 q^{36} + 12 q^{37} - 66 q^{38} - 42 q^{39} - 90 q^{40} - 12 q^{41} - 54 q^{43} + 18 q^{44} + 12 q^{45} + 30 q^{46} - 66 q^{47} + 90 q^{48} - 78 q^{49} - 12 q^{50} - 114 q^{51} + 78 q^{52} - 12 q^{53} + 114 q^{54} - 90 q^{55} + 6 q^{56} - 156 q^{57} + 30 q^{58} - 102 q^{59} + 102 q^{60} + 78 q^{62} - 72 q^{63} + 30 q^{64} - 60 q^{65} + 18 q^{66} - 66 q^{67} - 30 q^{68} - 24 q^{69} - 60 q^{70} - 126 q^{71} - 66 q^{72} - 24 q^{73} - 198 q^{74} - 150 q^{75} - 174 q^{76} - 18 q^{77} - 222 q^{78} - 66 q^{79} - 258 q^{80} - 192 q^{81} - 174 q^{82} - 186 q^{83} - 144 q^{84} - 84 q^{85} - 246 q^{86} - 282 q^{87} - 174 q^{88} - 156 q^{89} - 186 q^{90} - 108 q^{91} - 210 q^{92} - 264 q^{93} - 102 q^{94} - 186 q^{95} - 78 q^{96} - 240 q^{97} - 60 q^{98} - 120 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(392))\)

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
392.2.a \(\chi_{392}(1, \cdot)\) 392.2.a.a 1 1
392.2.a.b 1
392.2.a.c 1
392.2.a.d 1
392.2.a.e 1
392.2.a.f 1
392.2.a.g 2
392.2.a.h 2
392.2.b \(\chi_{392}(197, \cdot)\) 392.2.b.a 2 1
392.2.b.b 2
392.2.b.c 4
392.2.b.d 4
392.2.b.e 6
392.2.b.f 6
392.2.b.g 12
392.2.e \(\chi_{392}(195, \cdot)\) 392.2.e.a 4 1
392.2.e.b 4
392.2.e.c 8
392.2.e.d 8
392.2.e.e 12
392.2.f \(\chi_{392}(391, \cdot)\) None 0 1
392.2.i \(\chi_{392}(177, \cdot)\) 392.2.i.a 2 2
392.2.i.b 2
392.2.i.c 2
392.2.i.d 2
392.2.i.e 2
392.2.i.f 2
392.2.i.g 4
392.2.i.h 4
392.2.l \(\chi_{392}(31, \cdot)\) None 0 2
392.2.m \(\chi_{392}(19, \cdot)\) 392.2.m.a 4 2
392.2.m.b 8
392.2.m.c 8
392.2.m.d 8
392.2.m.e 8
392.2.m.f 8
392.2.m.g 12
392.2.m.h 16
392.2.p \(\chi_{392}(165, \cdot)\) 392.2.p.a 4 2
392.2.p.b 4
392.2.p.c 4
392.2.p.d 8
392.2.p.e 8
392.2.p.f 8
392.2.p.g 12
392.2.p.h 24
392.2.q \(\chi_{392}(57, \cdot)\) 392.2.q.a 42 6
392.2.q.b 42
392.2.t \(\chi_{392}(55, \cdot)\) None 0 6
392.2.u \(\chi_{392}(27, \cdot)\) 392.2.u.a 324 6
392.2.x \(\chi_{392}(29, \cdot)\) 392.2.x.a 324 6
392.2.y \(\chi_{392}(9, \cdot)\) 392.2.y.a 84 12
392.2.y.b 84
392.2.z \(\chi_{392}(37, \cdot)\) 392.2.z.a 648 12
392.2.bc \(\chi_{392}(3, \cdot)\) 392.2.bc.a 648 12
392.2.bd \(\chi_{392}(47, \cdot)\) None 0 12

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(392))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(392)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(392))\)\(^{\oplus 1}\)