Properties

Label 90.2.a
Level $90$
Weight $2$
Character orbit 90.a
Rep. character $\chi_{90}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
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: \(\Q\)
Newform subspaces: \( 3 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(90))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
90.2.a.a 90.a 1.a $1$ $0.719$ \(\Q\) None \(-1\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
90.2.a.b 90.a 1.a $1$ $0.719$ \(\Q\) None \(1\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
90.2.a.c 90.a 1.a $1$ $0.719$ \(\Q\) None \(1\) \(0\) \(1\) \(-4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)

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

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