Properties

Label 9.24.a.a
Level $9$
Weight $24$
Character orbit 9.a
Self dual yes
Analytic conductor $30.168$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.1683633611\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 1128 q^{2} - 7116224 q^{4} + 48863730 q^{5} - 1723688680 q^{7} + 17489450496 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 1128 q^{2} - 7116224 q^{4} + 48863730 q^{5} - 1723688680 q^{7} + 17489450496 q^{8} - 55118287440 q^{10} + 1428263180124 q^{11} - 8220964044826 q^{13} + 1944320831040 q^{14} + 39967113416704 q^{16} + 5989210330446 q^{17} + 680005481275676 q^{19} - 347725248155520 q^{20} - 16\!\cdots\!72 q^{22}+ \cdots + 27\!\cdots\!04 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1128.00 0 −7.11622e6 4.88637e7 0 −1.72369e9 1.74895e10 0 −5.51183e10
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.24.a.a 1
3.b odd 2 1 3.24.a.a 1
12.b even 2 1 48.24.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.24.a.a 1 3.b odd 2 1
9.24.a.a 1 1.a even 1 1 trivial
48.24.a.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1128 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1128 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 48863730 \) Copy content Toggle raw display
$7$ \( T + 1723688680 \) Copy content Toggle raw display
$11$ \( T - 1428263180124 \) Copy content Toggle raw display
$13$ \( T + 8220964044826 \) Copy content Toggle raw display
$17$ \( T - 5989210330446 \) Copy content Toggle raw display
$19$ \( T - 680005481275676 \) Copy content Toggle raw display
$23$ \( T + 15440648191080 \) Copy content Toggle raw display
$29$ \( T + 11\!\cdots\!22 \) Copy content Toggle raw display
$31$ \( T + 90\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T + 12\!\cdots\!70 \) Copy content Toggle raw display
$41$ \( T + 52\!\cdots\!30 \) Copy content Toggle raw display
$43$ \( T + 24\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T - 23\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T - 44\!\cdots\!50 \) Copy content Toggle raw display
$59$ \( T - 32\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T + 19\!\cdots\!22 \) Copy content Toggle raw display
$67$ \( T + 64\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T + 35\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T - 33\!\cdots\!70 \) Copy content Toggle raw display
$79$ \( T + 68\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T - 11\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T - 23\!\cdots\!74 \) Copy content Toggle raw display
$97$ \( T + 30\!\cdots\!86 \) Copy content Toggle raw display
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