Properties

Label 81.2.a
Level $81$
Weight $2$
Character orbit 81.a
Rep. character $\chi_{81}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $1$
Sturm bound $18$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(81))\).

Total New Old
Modular forms 15 6 9
Cusp forms 4 2 2
Eisenstein series 11 4 7

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.
\(-\)\(2\)

Trace form

\( 2 q + 2 q^{4} + 4 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{4} + 4 q^{7} - 6 q^{10} - 2 q^{13} - 10 q^{16} + 4 q^{19} - 12 q^{22} - 4 q^{25} + 4 q^{28} + 16 q^{31} + 18 q^{34} - 14 q^{37} + 6 q^{40} + 4 q^{43} + 12 q^{46} - 6 q^{49} - 2 q^{52} + 12 q^{55} + 6 q^{58} - 14 q^{61} + 2 q^{64} - 20 q^{67} - 12 q^{70} - 14 q^{73} + 4 q^{76} + 4 q^{79} - 24 q^{82} - 18 q^{85} + 12 q^{88} - 4 q^{91} + 24 q^{94} + 4 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(81))\) 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
81.2.a.a 81.a 1.a $2$ $0.647$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-\beta q^{5}+2q^{7}-\beta q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(81))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(81)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)