Properties

Label 9.4.a
Level 9
Weight 4
Character orbit 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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
9.4.a.a \(1\) \(0.531\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(20\) \(+\) \(q-8q^{4}+20q^{7}-70q^{13}+2^{6}q^{16}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T^{2} \)
$3$ 1
$5$ \( 1 + 125 T^{2} \)
$7$ \( 1 - 20 T + 343 T^{2} \)
$11$ \( 1 + 1331 T^{2} \)
$13$ \( 1 + 70 T + 2197 T^{2} \)
$17$ \( 1 + 4913 T^{2} \)
$19$ \( 1 - 56 T + 6859 T^{2} \)
$23$ \( 1 + 12167 T^{2} \)
$29$ \( 1 + 24389 T^{2} \)
$31$ \( 1 - 308 T + 29791 T^{2} \)
$37$ \( 1 - 110 T + 50653 T^{2} \)
$41$ \( 1 + 68921 T^{2} \)
$43$ \( 1 + 520 T + 79507 T^{2} \)
$47$ \( 1 + 103823 T^{2} \)
$53$ \( 1 + 148877 T^{2} \)
$59$ \( 1 + 205379 T^{2} \)
$61$ \( 1 - 182 T + 226981 T^{2} \)
$67$ \( 1 + 880 T + 300763 T^{2} \)
$71$ \( 1 + 357911 T^{2} \)
$73$ \( 1 - 1190 T + 389017 T^{2} \)
$79$ \( 1 - 884 T + 493039 T^{2} \)
$83$ \( 1 + 571787 T^{2} \)
$89$ \( 1 + 704969 T^{2} \)
$97$ \( 1 + 1330 T + 912673 T^{2} \)
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