Properties

Label 56.4
Level 56
Weight 4
Dimension 144
Nonzero newspaces 6
Newform subspaces 12
Sturm bound 768
Trace bound 2

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Defining parameters

Level: \( N \) = \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 12 \)
Sturm bound: \(768\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(56))\).

Total New Old
Modular forms 324 164 160
Cusp forms 252 144 108
Eisenstein series 72 20 52

Trace form

\( 144 q - 2 q^{2} + 2 q^{3} + 18 q^{4} + 4 q^{5} - 62 q^{6} - 14 q^{7} - 92 q^{8} - 28 q^{9} + O(q^{10}) \) \( 144 q - 2 q^{2} + 2 q^{3} + 18 q^{4} + 4 q^{5} - 62 q^{6} - 14 q^{7} - 92 q^{8} - 28 q^{9} + 106 q^{10} + 124 q^{11} + 106 q^{12} + 34 q^{13} - 22 q^{14} - 120 q^{15} - 38 q^{16} - 50 q^{17} + 152 q^{18} + 74 q^{19} + 292 q^{20} - 72 q^{21} + 246 q^{22} + 588 q^{23} - 322 q^{24} - 200 q^{25} - 1316 q^{26} - 952 q^{27} - 1398 q^{28} - 396 q^{29} - 790 q^{30} - 1356 q^{31} + 488 q^{32} - 1018 q^{33} + 478 q^{34} + 546 q^{35} + 1314 q^{36} + 954 q^{37} + 1880 q^{38} + 2712 q^{39} + 296 q^{40} + 1648 q^{41} - 1078 q^{42} + 1564 q^{43} - 2190 q^{44} + 1642 q^{45} - 1664 q^{46} - 1596 q^{47} - 1410 q^{48} + 928 q^{49} + 1090 q^{50} - 2588 q^{51} + 3436 q^{52} - 1490 q^{53} + 3038 q^{54} - 3848 q^{55} + 2624 q^{56} - 3408 q^{57} + 3780 q^{58} - 1226 q^{59} + 4440 q^{60} - 1604 q^{61} + 1592 q^{62} + 950 q^{63} - 1878 q^{64} - 652 q^{65} - 1674 q^{66} + 5036 q^{67} + 1338 q^{68} + 5780 q^{69} + 578 q^{70} + 4472 q^{71} + 236 q^{72} + 2386 q^{73} - 1994 q^{74} + 118 q^{75} - 2734 q^{76} - 186 q^{77} - 5584 q^{78} + 12 q^{79} - 5412 q^{80} - 666 q^{81} - 6118 q^{82} - 7666 q^{83} - 9418 q^{84} - 6184 q^{85} - 3674 q^{86} - 7188 q^{87} - 5214 q^{88} - 3158 q^{89} - 6088 q^{90} - 3066 q^{91} - 6198 q^{92} + 3130 q^{93} - 3738 q^{94} + 772 q^{95} - 718 q^{96} + 4072 q^{97} + 4150 q^{98} + 6532 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(56))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
56.4.a \(\chi_{56}(1, \cdot)\) 56.4.a.a 1 1
56.4.a.b 1
56.4.a.c 2
56.4.b \(\chi_{56}(29, \cdot)\) 56.4.b.a 8 1
56.4.b.b 10
56.4.e \(\chi_{56}(27, \cdot)\) 56.4.e.a 2 1
56.4.e.b 4
56.4.e.c 16
56.4.f \(\chi_{56}(55, \cdot)\) None 0 1
56.4.i \(\chi_{56}(9, \cdot)\) 56.4.i.a 6 2
56.4.i.b 6
56.4.l \(\chi_{56}(31, \cdot)\) None 0 2
56.4.m \(\chi_{56}(3, \cdot)\) 56.4.m.a 44 2
56.4.p \(\chi_{56}(37, \cdot)\) 56.4.p.a 44 2

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces

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