Properties

Label 90.2
Level 90
Weight 2
Dimension 53
Nonzero newspaces 6
Newforms 12
Sturm bound 864
Trace bound 1

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newforms: \( 12 \)
Sturm bound: \(864\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(90))\).

Total New Old
Modular forms 280 53 227
Cusp forms 153 53 100
Eisenstein series 127 0 127

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(90))\)

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
90.2.a \(\chi_{90}(1, \cdot)\) 90.2.a.a 1 1
90.2.a.b 1
90.2.a.c 1
90.2.c \(\chi_{90}(19, \cdot)\) 90.2.c.a 2 1
90.2.e \(\chi_{90}(31, \cdot)\) 90.2.e.a 2 2
90.2.e.b 2
90.2.e.c 4
90.2.f \(\chi_{90}(17, \cdot)\) 90.2.f.a 4 2
90.2.i \(\chi_{90}(49, \cdot)\) 90.2.i.a 4 2
90.2.i.b 8
90.2.l \(\chi_{90}(23, \cdot)\) 90.2.l.a 8 4
90.2.l.b 16

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(90))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(90)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)