Properties

Label 576.4.a.d
Level $576$
Weight $4$
Character orbit 576.a
Self dual yes
Analytic conductor $33.985$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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 72)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 16 q^{5} + 12 q^{7} + O(q^{10}) \) \( q - 16 q^{5} + 12 q^{7} + 64 q^{11} - 58 q^{13} + 32 q^{17} - 136 q^{19} + 128 q^{23} + 131 q^{25} + 144 q^{29} - 20 q^{31} - 192 q^{35} + 18 q^{37} - 288 q^{41} - 200 q^{43} - 384 q^{47} - 199 q^{49} - 496 q^{53} - 1024 q^{55} - 128 q^{59} + 458 q^{61} + 928 q^{65} - 496 q^{67} - 512 q^{71} - 602 q^{73} + 768 q^{77} - 1108 q^{79} + 704 q^{83} - 512 q^{85} - 960 q^{89} - 696 q^{91} + 2176 q^{95} + 206 q^{97} + O(q^{100}) \)

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 −16.0000 0 12.0000 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

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 576.4.a.d 1
3.b odd 2 1 576.4.a.x 1
4.b odd 2 1 576.4.a.c 1
8.b even 2 1 144.4.a.f 1
8.d odd 2 1 72.4.a.d yes 1
12.b even 2 1 576.4.a.w 1
24.f even 2 1 72.4.a.a 1
24.h odd 2 1 144.4.a.a 1
40.e odd 2 1 1800.4.a.ba 1
40.k even 4 2 1800.4.f.x 2
72.l even 6 2 648.4.i.l 2
72.p odd 6 2 648.4.i.a 2
120.m even 2 1 1800.4.a.z 1
120.q odd 4 2 1800.4.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.a.a 1 24.f even 2 1
72.4.a.d yes 1 8.d odd 2 1
144.4.a.a 1 24.h odd 2 1
144.4.a.f 1 8.b even 2 1
576.4.a.c 1 4.b odd 2 1
576.4.a.d 1 1.a even 1 1 trivial
576.4.a.w 1 12.b even 2 1
576.4.a.x 1 3.b odd 2 1
648.4.i.a 2 72.p odd 6 2
648.4.i.l 2 72.l even 6 2
1800.4.a.z 1 120.m even 2 1
1800.4.a.ba 1 40.e odd 2 1
1800.4.f.b 2 120.q odd 4 2
1800.4.f.x 2 40.k even 4 2

Hecke kernels

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

\( T_{5} + 16 \)
\( T_{7} - 12 \)
\( T_{11} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 16 + T \)
$7$ \( -12 + T \)
$11$ \( -64 + T \)
$13$ \( 58 + T \)
$17$ \( -32 + T \)
$19$ \( 136 + T \)
$23$ \( -128 + T \)
$29$ \( -144 + T \)
$31$ \( 20 + T \)
$37$ \( -18 + T \)
$41$ \( 288 + T \)
$43$ \( 200 + T \)
$47$ \( 384 + T \)
$53$ \( 496 + T \)
$59$ \( 128 + T \)
$61$ \( -458 + T \)
$67$ \( 496 + T \)
$71$ \( 512 + T \)
$73$ \( 602 + T \)
$79$ \( 1108 + T \)
$83$ \( -704 + T \)
$89$ \( 960 + T \)
$97$ \( -206 + T \)
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