Properties

Label 576.4.a.p
Level $576$
Weight $4$
Character orbit 576.a
Self dual yes
Analytic conductor $33.985$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 576.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.9851001633\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 4 q^{5} + O(q^{10}) \) \( q + 4 q^{5} - 18 q^{13} - 104 q^{17} - 109 q^{25} + 284 q^{29} - 214 q^{37} - 472 q^{41} - 343 q^{49} + 572 q^{53} - 830 q^{61} - 72 q^{65} - 1098 q^{73} - 416 q^{85} + 176 q^{89} - 594 q^{97} + O(q^{100}) \)

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
0 0 0 4.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.4.a.p 1
3.b odd 2 1 576.4.a.i 1
4.b odd 2 1 CM 576.4.a.p 1
8.b even 2 1 288.4.a.d 1
8.d odd 2 1 288.4.a.d 1
12.b even 2 1 576.4.a.i 1
24.f even 2 1 288.4.a.g yes 1
24.h odd 2 1 288.4.a.g yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.4.a.d 1 8.b even 2 1
288.4.a.d 1 8.d odd 2 1
288.4.a.g yes 1 24.f even 2 1
288.4.a.g yes 1 24.h odd 2 1
576.4.a.i 1 3.b odd 2 1
576.4.a.i 1 12.b even 2 1
576.4.a.p 1 1.a even 1 1 trivial
576.4.a.p 1 4.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(576))\):

\( T_{5} - 4 \)
\( T_{7} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -4 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 18 + T \)
$17$ \( 104 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -284 + T \)
$31$ \( T \)
$37$ \( 214 + T \)
$41$ \( 472 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -572 + T \)
$59$ \( T \)
$61$ \( 830 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( 1098 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -176 + T \)
$97$ \( 594 + T \)
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