Properties

Label 9.8.a
Level $9$
Weight $8$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $2$
Sturm bound $8$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(9))\).

Total New Old
Modular forms 9 4 5
Cusp forms 5 3 2
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
\(+\)\(2\)
\(-\)\(1\)

Trace form

\( 3 q - 6 q^{2} + 372 q^{4} - 390 q^{5} + 456 q^{7} + 1320 q^{8} - 9180 q^{10} + 948 q^{11} + 8682 q^{13} + 384 q^{14} + 19344 q^{16} - 28386 q^{17} + 57732 q^{19} + 35880 q^{20} - 236088 q^{22} + 15288 q^{23}+ \cdots + 4916682 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
9.8.a.a 9.a 1.a $1$ $2.811$ \(\Q\) None 3.8.a.a \(-6\) \(0\) \(-390\) \(-64\) $-$ $\mathrm{SU}(2)$ \(q-6q^{2}-92q^{4}-390q^{5}-2^{6}q^{7}+\cdots\)
9.8.a.b 9.a 1.a $2$ $2.811$ \(\Q(\sqrt{10}) \) None 9.8.a.b \(0\) \(0\) \(0\) \(520\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+232q^{4}-2^{4}\beta q^{5}+260q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(9)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)