Properties

Label 75.2
Level 75
Weight 2
Dimension 119
Nonzero newspaces 6
Newforms 12
Sturm bound 800
Trace bound 1

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newforms: \( 12 \)
Sturm bound: \(800\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(75))\).

Total New Old
Modular forms 256 159 97
Cusp forms 145 119 26
Eisenstein series 111 40 71

Trace form

\(119q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 25q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 23q^{6} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut -\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(119q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 25q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 23q^{6} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut -\mathstrut 13q^{9} \) \(\mathstrut -\mathstrut 34q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 19q^{12} \) \(\mathstrut -\mathstrut 30q^{13} \) \(\mathstrut -\mathstrut 24q^{14} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut -\mathstrut 33q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut q^{18} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 36q^{20} \) \(\mathstrut -\mathstrut 6q^{21} \) \(\mathstrut +\mathstrut 16q^{22} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 51q^{24} \) \(\mathstrut +\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 6q^{26} \) \(\mathstrut -\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 44q^{28} \) \(\mathstrut +\mathstrut 14q^{29} \) \(\mathstrut +\mathstrut 14q^{30} \) \(\mathstrut -\mathstrut 28q^{31} \) \(\mathstrut +\mathstrut 17q^{32} \) \(\mathstrut -\mathstrut 10q^{33} \) \(\mathstrut -\mathstrut 20q^{34} \) \(\mathstrut -\mathstrut 20q^{35} \) \(\mathstrut +\mathstrut 9q^{36} \) \(\mathstrut -\mathstrut 28q^{37} \) \(\mathstrut +\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 12q^{39} \) \(\mathstrut +\mathstrut 2q^{40} \) \(\mathstrut -\mathstrut 22q^{41} \) \(\mathstrut +\mathstrut 66q^{42} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 48q^{44} \) \(\mathstrut +\mathstrut 64q^{45} \) \(\mathstrut +\mathstrut 12q^{46} \) \(\mathstrut +\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 67q^{48} \) \(\mathstrut +\mathstrut 47q^{49} \) \(\mathstrut +\mathstrut 106q^{50} \) \(\mathstrut +\mathstrut 18q^{51} \) \(\mathstrut +\mathstrut 98q^{52} \) \(\mathstrut +\mathstrut 56q^{53} \) \(\mathstrut +\mathstrut 115q^{54} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 58q^{57} \) \(\mathstrut +\mathstrut 26q^{58} \) \(\mathstrut +\mathstrut 28q^{59} \) \(\mathstrut +\mathstrut 14q^{60} \) \(\mathstrut -\mathstrut 54q^{61} \) \(\mathstrut +\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 28q^{63} \) \(\mathstrut -\mathstrut 101q^{64} \) \(\mathstrut -\mathstrut 58q^{65} \) \(\mathstrut -\mathstrut 58q^{66} \) \(\mathstrut -\mathstrut 112q^{67} \) \(\mathstrut -\mathstrut 122q^{68} \) \(\mathstrut -\mathstrut 104q^{69} \) \(\mathstrut -\mathstrut 140q^{70} \) \(\mathstrut -\mathstrut 56q^{71} \) \(\mathstrut -\mathstrut 171q^{72} \) \(\mathstrut -\mathstrut 114q^{73} \) \(\mathstrut -\mathstrut 134q^{74} \) \(\mathstrut -\mathstrut 114q^{75} \) \(\mathstrut -\mathstrut 124q^{76} \) \(\mathstrut -\mathstrut 96q^{77} \) \(\mathstrut -\mathstrut 118q^{78} \) \(\mathstrut -\mathstrut 100q^{79} \) \(\mathstrut -\mathstrut 154q^{80} \) \(\mathstrut -\mathstrut 61q^{81} \) \(\mathstrut -\mathstrut 46q^{82} \) \(\mathstrut -\mathstrut 28q^{83} \) \(\mathstrut -\mathstrut 106q^{84} \) \(\mathstrut +\mathstrut 38q^{85} \) \(\mathstrut -\mathstrut 4q^{86} \) \(\mathstrut -\mathstrut 14q^{87} \) \(\mathstrut +\mathstrut 64q^{88} \) \(\mathstrut +\mathstrut 72q^{89} \) \(\mathstrut -\mathstrut 24q^{90} \) \(\mathstrut +\mathstrut 12q^{91} \) \(\mathstrut +\mathstrut 32q^{92} \) \(\mathstrut +\mathstrut 68q^{93} \) \(\mathstrut +\mathstrut 172q^{94} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut 33q^{96} \) \(\mathstrut +\mathstrut 158q^{97} \) \(\mathstrut +\mathstrut 95q^{98} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(75))\)

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
75.2.a \(\chi_{75}(1, \cdot)\) 75.2.a.a 1 1
75.2.a.b 1
75.2.a.c 1
75.2.b \(\chi_{75}(49, \cdot)\) 75.2.b.a 2 1
75.2.b.b 2
75.2.e \(\chi_{75}(32, \cdot)\) 75.2.e.a 4 2
75.2.e.b 4
75.2.g \(\chi_{75}(16, \cdot)\) 75.2.g.a 4 4
75.2.g.b 8
75.2.g.c 12
75.2.i \(\chi_{75}(4, \cdot)\) 75.2.i.a 16 4
75.2.l \(\chi_{75}(2, \cdot)\) 75.2.l.a 64 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)