Properties

Label 9.4.a
Level $9$
Weight $4$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $4$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(9))\).

Total New Old
Modular forms 5 2 3
Cusp forms 1 1 0
Eisenstein series 4 1 3

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

\(3\)Dim.
\(+\)\(1\)

Trace form

\( q - 8 q^{4} + 20 q^{7} + O(q^{10}) \) \( q - 8 q^{4} + 20 q^{7} - 70 q^{13} + 64 q^{16} + 56 q^{19} - 125 q^{25} - 160 q^{28} + 308 q^{31} + 110 q^{37} - 520 q^{43} + 57 q^{49} + 560 q^{52} + 182 q^{61} - 512 q^{64} - 880 q^{67} + 1190 q^{73} - 448 q^{76} + 884 q^{79} - 1400 q^{91} - 1330 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
9.4.a.a 9.a 1.a $1$ $0.531$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(20\) $+$ $N(\mathrm{U}(1))$ \(q-8q^{4}+20q^{7}-70q^{13}+2^{6}q^{16}+\cdots\)