Properties

Label 56.2
Level 56
Weight 2
Dimension 42
Nonzero newspaces 6
Newforms 10
Sturm bound 384
Trace bound 2

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Defining parameters

Level: \( N \) = \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newforms: \( 10 \)
Sturm bound: \(384\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(56))\).

Total New Old
Modular forms 132 62 70
Cusp forms 61 42 19
Eisenstein series 71 20 51

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)

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
56.2.a \(\chi_{56}(1, \cdot)\) 56.2.a.a 1 1
56.2.a.b 1
56.2.b \(\chi_{56}(29, \cdot)\) 56.2.b.a 2 1
56.2.b.b 4
56.2.e \(\chi_{56}(27, \cdot)\) 56.2.e.a 2 1
56.2.e.b 4
56.2.f \(\chi_{56}(55, \cdot)\) None 0 1
56.2.i \(\chi_{56}(9, \cdot)\) 56.2.i.a 2 2
56.2.i.b 2
56.2.l \(\chi_{56}(31, \cdot)\) None 0 2
56.2.m \(\chi_{56}(3, \cdot)\) 56.2.m.a 12 2
56.2.p \(\chi_{56}(37, \cdot)\) 56.2.p.a 12 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces

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