Properties

Label 51.2
Level 51
Weight 2
Dimension 55
Nonzero newspaces 5
Newforms 7
Sturm bound 384
Trace bound 4

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Defining parameters

Level: \( N \) = \( 51 = 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 5 \)
Newforms: \( 7 \)
Sturm bound: \(384\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(51))\).

Total New Old
Modular forms 128 87 41
Cusp forms 65 55 10
Eisenstein series 63 32 31

Trace form

\(55q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 23q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 11q^{6} \) \(\mathstrut -\mathstrut 24q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut -\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(55q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 23q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 11q^{6} \) \(\mathstrut -\mathstrut 24q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut -\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 26q^{10} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 9q^{12} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut +\mathstrut 8q^{14} \) \(\mathstrut +\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 25q^{16} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut +\mathstrut 13q^{18} \) \(\mathstrut -\mathstrut 20q^{19} \) \(\mathstrut +\mathstrut 14q^{20} \) \(\mathstrut +\mathstrut 8q^{21} \) \(\mathstrut -\mathstrut 20q^{22} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 17q^{24} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut +\mathstrut 30q^{26} \) \(\mathstrut -\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 24q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 38q^{30} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 17q^{32} \) \(\mathstrut +\mathstrut 20q^{33} \) \(\mathstrut +\mathstrut 61q^{34} \) \(\mathstrut +\mathstrut 16q^{35} \) \(\mathstrut +\mathstrut 17q^{36} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 20q^{38} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 22q^{40} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 32q^{42} \) \(\mathstrut -\mathstrut 12q^{43} \) \(\mathstrut -\mathstrut 20q^{44} \) \(\mathstrut -\mathstrut 38q^{45} \) \(\mathstrut -\mathstrut 56q^{46} \) \(\mathstrut -\mathstrut 48q^{47} \) \(\mathstrut -\mathstrut 95q^{48} \) \(\mathstrut -\mathstrut 73q^{49} \) \(\mathstrut -\mathstrut 93q^{50} \) \(\mathstrut -\mathstrut 57q^{51} \) \(\mathstrut -\mathstrut 130q^{52} \) \(\mathstrut -\mathstrut 30q^{53} \) \(\mathstrut -\mathstrut 27q^{54} \) \(\mathstrut -\mathstrut 24q^{55} \) \(\mathstrut -\mathstrut 40q^{56} \) \(\mathstrut -\mathstrut 12q^{57} \) \(\mathstrut +\mathstrut 6q^{58} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut -\mathstrut 10q^{60} \) \(\mathstrut +\mathstrut 18q^{61} \) \(\mathstrut +\mathstrut 16q^{62} \) \(\mathstrut +\mathstrut 56q^{63} \) \(\mathstrut +\mathstrut 17q^{64} \) \(\mathstrut +\mathstrut 36q^{65} \) \(\mathstrut +\mathstrut 36q^{66} \) \(\mathstrut -\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 81q^{68} \) \(\mathstrut +\mathstrut 72q^{69} \) \(\mathstrut +\mathstrut 112q^{70} \) \(\mathstrut +\mathstrut 24q^{71} \) \(\mathstrut +\mathstrut 73q^{72} \) \(\mathstrut +\mathstrut 110q^{73} \) \(\mathstrut +\mathstrut 54q^{74} \) \(\mathstrut +\mathstrut 89q^{75} \) \(\mathstrut +\mathstrut 84q^{76} \) \(\mathstrut +\mathstrut 64q^{77} \) \(\mathstrut +\mathstrut 118q^{78} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut +\mathstrut 86q^{80} \) \(\mathstrut +\mathstrut 47q^{81} \) \(\mathstrut +\mathstrut 74q^{82} \) \(\mathstrut +\mathstrut 28q^{83} \) \(\mathstrut +\mathstrut 40q^{84} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut -\mathstrut 68q^{86} \) \(\mathstrut -\mathstrut 38q^{87} \) \(\mathstrut -\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 42q^{89} \) \(\mathstrut -\mathstrut 98q^{90} \) \(\mathstrut -\mathstrut 32q^{91} \) \(\mathstrut -\mathstrut 8q^{92} \) \(\mathstrut -\mathstrut 72q^{93} \) \(\mathstrut -\mathstrut 32q^{94} \) \(\mathstrut -\mathstrut 56q^{95} \) \(\mathstrut -\mathstrut 111q^{96} \) \(\mathstrut -\mathstrut 50q^{97} \) \(\mathstrut -\mathstrut 99q^{98} \) \(\mathstrut -\mathstrut 60q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(51))\)

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
51.2.a \(\chi_{51}(1, \cdot)\) 51.2.a.a 1 1
51.2.a.b 2
51.2.d \(\chi_{51}(16, \cdot)\) 51.2.d.a 2 1
51.2.d.b 2
51.2.e \(\chi_{51}(4, \cdot)\) 51.2.e.a 8 2
51.2.h \(\chi_{51}(19, \cdot)\) 51.2.h.a 8 4
51.2.i \(\chi_{51}(5, \cdot)\) 51.2.i.a 32 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(51))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(51)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 2}\)