Properties

Label 135.2
Level 135
Weight 2
Dimension 422
Nonzero newspaces 9
Newforms 16
Sturm bound 2592
Trace bound 4

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Defining parameters

Level: \( N \) = \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 9 \)
Newforms: \( 16 \)
Sturm bound: \(2592\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(135))\).

Total New Old
Modular forms 768 518 250
Cusp forms 529 422 107
Eisenstein series 239 96 143

Trace form

\(422q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut -\mathstrut 48q^{6} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 36q^{8} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(422q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 13q^{5} \) \(\mathstrut -\mathstrut 48q^{6} \) \(\mathstrut -\mathstrut 14q^{7} \) \(\mathstrut -\mathstrut 36q^{8} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut -\mathstrut 33q^{10} \) \(\mathstrut -\mathstrut 46q^{11} \) \(\mathstrut -\mathstrut 48q^{12} \) \(\mathstrut -\mathstrut 34q^{13} \) \(\mathstrut -\mathstrut 66q^{14} \) \(\mathstrut -\mathstrut 33q^{15} \) \(\mathstrut -\mathstrut 80q^{16} \) \(\mathstrut -\mathstrut 46q^{17} \) \(\mathstrut -\mathstrut 42q^{18} \) \(\mathstrut -\mathstrut 16q^{19} \) \(\mathstrut -\mathstrut 45q^{20} \) \(\mathstrut -\mathstrut 24q^{21} \) \(\mathstrut -\mathstrut 58q^{22} \) \(\mathstrut -\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 12q^{24} \) \(\mathstrut -\mathstrut 37q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut -\mathstrut 6q^{27} \) \(\mathstrut -\mathstrut 60q^{28} \) \(\mathstrut -\mathstrut 32q^{29} \) \(\mathstrut -\mathstrut 15q^{30} \) \(\mathstrut -\mathstrut 70q^{31} \) \(\mathstrut +\mathstrut 16q^{32} \) \(\mathstrut -\mathstrut 12q^{33} \) \(\mathstrut -\mathstrut 46q^{34} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut -\mathstrut 12q^{36} \) \(\mathstrut -\mathstrut 36q^{37} \) \(\mathstrut -\mathstrut 16q^{38} \) \(\mathstrut -\mathstrut 30q^{39} \) \(\mathstrut +\mathstrut 31q^{40} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut +\mathstrut 36q^{42} \) \(\mathstrut +\mathstrut 14q^{43} \) \(\mathstrut +\mathstrut 166q^{44} \) \(\mathstrut +\mathstrut 45q^{45} \) \(\mathstrut -\mathstrut 10q^{46} \) \(\mathstrut +\mathstrut 98q^{47} \) \(\mathstrut +\mathstrut 126q^{48} \) \(\mathstrut +\mathstrut 152q^{50} \) \(\mathstrut +\mathstrut 36q^{51} \) \(\mathstrut +\mathstrut 30q^{52} \) \(\mathstrut +\mathstrut 80q^{53} \) \(\mathstrut +\mathstrut 156q^{54} \) \(\mathstrut -\mathstrut 60q^{55} \) \(\mathstrut +\mathstrut 66q^{56} \) \(\mathstrut +\mathstrut 18q^{57} \) \(\mathstrut -\mathstrut 70q^{58} \) \(\mathstrut -\mathstrut 28q^{59} \) \(\mathstrut +\mathstrut 48q^{60} \) \(\mathstrut -\mathstrut 126q^{61} \) \(\mathstrut -\mathstrut 84q^{62} \) \(\mathstrut -\mathstrut 42q^{63} \) \(\mathstrut -\mathstrut 84q^{64} \) \(\mathstrut -\mathstrut 87q^{65} \) \(\mathstrut -\mathstrut 30q^{66} \) \(\mathstrut -\mathstrut 56q^{67} \) \(\mathstrut -\mathstrut 62q^{68} \) \(\mathstrut -\mathstrut 42q^{69} \) \(\mathstrut -\mathstrut 45q^{70} \) \(\mathstrut -\mathstrut 158q^{71} \) \(\mathstrut +\mathstrut 36q^{72} \) \(\mathstrut -\mathstrut 34q^{73} \) \(\mathstrut -\mathstrut 14q^{74} \) \(\mathstrut -\mathstrut 21q^{75} \) \(\mathstrut +\mathstrut 28q^{76} \) \(\mathstrut +\mathstrut 54q^{77} \) \(\mathstrut +\mathstrut 96q^{78} \) \(\mathstrut +\mathstrut 94q^{79} \) \(\mathstrut +\mathstrut 150q^{80} \) \(\mathstrut -\mathstrut 24q^{81} \) \(\mathstrut +\mathstrut 120q^{82} \) \(\mathstrut +\mathstrut 78q^{83} \) \(\mathstrut +\mathstrut 192q^{84} \) \(\mathstrut +\mathstrut 63q^{85} \) \(\mathstrut +\mathstrut 170q^{86} \) \(\mathstrut +\mathstrut 78q^{87} \) \(\mathstrut +\mathstrut 168q^{88} \) \(\mathstrut +\mathstrut 186q^{89} \) \(\mathstrut +\mathstrut 186q^{90} \) \(\mathstrut +\mathstrut 38q^{91} \) \(\mathstrut +\mathstrut 402q^{92} \) \(\mathstrut +\mathstrut 198q^{93} \) \(\mathstrut +\mathstrut 146q^{94} \) \(\mathstrut +\mathstrut 165q^{95} \) \(\mathstrut +\mathstrut 168q^{96} \) \(\mathstrut +\mathstrut 60q^{97} \) \(\mathstrut +\mathstrut 370q^{98} \) \(\mathstrut +\mathstrut 150q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(135))\)

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
135.2.a \(\chi_{135}(1, \cdot)\) 135.2.a.a 1 1
135.2.a.b 1
135.2.a.c 2
135.2.a.d 2
135.2.b \(\chi_{135}(109, \cdot)\) 135.2.b.a 4 1
135.2.b.b 4
135.2.e \(\chi_{135}(46, \cdot)\) 135.2.e.a 2 2
135.2.e.b 6
135.2.f \(\chi_{135}(53, \cdot)\) 135.2.f.a 8 2
135.2.f.b 8
135.2.j \(\chi_{135}(19, \cdot)\) 135.2.j.a 8 2
135.2.k \(\chi_{135}(16, \cdot)\) 135.2.k.a 30 6
135.2.k.b 42
135.2.m \(\chi_{135}(8, \cdot)\) 135.2.m.a 16 4
135.2.p \(\chi_{135}(4, \cdot)\) 135.2.p.a 96 6
135.2.q \(\chi_{135}(2, \cdot)\) 135.2.q.a 192 12

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(135))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(135)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)