Properties

Label 90.2.a
Level 90
Weight 2
Character orbit a
Rep. character \(\chi_{90}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 3
Sturm bound 36
Trace bound 5

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Defining parameters

Level: \( N \) = \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 90.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(36\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(90))\).

Total New Old
Modular forms 26 3 23
Cusp forms 11 3 8
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\(3q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut q^{8} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 3q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut q^{20} \) \(\mathstrut -\mathstrut 12q^{22} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut +\mathstrut q^{32} \) \(\mathstrut +\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 18q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut +\mathstrut 6q^{41} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 3q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut -\mathstrut 6q^{52} \) \(\mathstrut +\mathstrut 6q^{53} \) \(\mathstrut +\mathstrut 12q^{55} \) \(\mathstrut -\mathstrut 4q^{56} \) \(\mathstrut +\mathstrut 18q^{58} \) \(\mathstrut -\mathstrut 6q^{61} \) \(\mathstrut +\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 3q^{64} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 12q^{67} \) \(\mathstrut -\mathstrut 6q^{68} \) \(\mathstrut -\mathstrut 8q^{70} \) \(\mathstrut -\mathstrut 18q^{73} \) \(\mathstrut +\mathstrut 2q^{74} \) \(\mathstrut -\mathstrut 12q^{76} \) \(\mathstrut +\mathstrut q^{80} \) \(\mathstrut +\mathstrut 6q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut -\mathstrut 18q^{85} \) \(\mathstrut -\mathstrut 4q^{86} \) \(\mathstrut -\mathstrut 12q^{88} \) \(\mathstrut -\mathstrut 18q^{89} \) \(\mathstrut -\mathstrut 24q^{91} \) \(\mathstrut -\mathstrut 4q^{95} \) \(\mathstrut +\mathstrut 6q^{97} \) \(\mathstrut +\mathstrut 9q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(90))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
90.2.a.a \(1\) \(0.719\) \(\Q\) None \(-1\) \(0\) \(1\) \(2\) \(+\) \(+\) \(-\) \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
90.2.a.b \(1\) \(0.719\) \(\Q\) None \(1\) \(0\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
90.2.a.c \(1\) \(0.719\) \(\Q\) None \(1\) \(0\) \(1\) \(-4\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(90)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)