Properties

Label 588.4.a.e
Level $588$
Weight $4$
Character orbit 588.a
Self dual yes
Analytic conductor $34.693$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.6931230834\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 4q^{5} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 4q^{5} + 9q^{9} - 20q^{11} + 4q^{13} - 12q^{15} - 24q^{17} - 44q^{19} + 72q^{23} - 109q^{25} + 27q^{27} - 38q^{29} - 184q^{31} - 60q^{33} - 30q^{37} + 12q^{39} + 216q^{41} - 164q^{43} - 36q^{45} - 520q^{47} - 72q^{51} - 146q^{53} + 80q^{55} - 132q^{57} - 460q^{59} - 628q^{61} - 16q^{65} + 556q^{67} + 216q^{69} + 592q^{71} - 1024q^{73} - 327q^{75} - 104q^{79} + 81q^{81} + 324q^{83} + 96q^{85} - 114q^{87} - 896q^{89} - 552q^{93} + 176q^{95} + 920q^{97} - 180q^{99} + O(q^{100}) \)

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 3.00000 0 −4.00000 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

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 588.4.a.e yes 1
3.b odd 2 1 1764.4.a.i 1
4.b odd 2 1 2352.4.a.g 1
7.b odd 2 1 588.4.a.b 1
7.c even 3 2 588.4.i.b 2
7.d odd 6 2 588.4.i.g 2
21.c even 2 1 1764.4.a.d 1
21.g even 6 2 1764.4.k.j 2
21.h odd 6 2 1764.4.k.g 2
28.d even 2 1 2352.4.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 7.b odd 2 1
588.4.a.e yes 1 1.a even 1 1 trivial
588.4.i.b 2 7.c even 3 2
588.4.i.g 2 7.d odd 6 2
1764.4.a.d 1 21.c even 2 1
1764.4.a.i 1 3.b odd 2 1
1764.4.k.g 2 21.h odd 6 2
1764.4.k.j 2 21.g even 6 2
2352.4.a.g 1 4.b odd 2 1
2352.4.a.be 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 4 + T \)
$7$ \( T \)
$11$ \( 20 + T \)
$13$ \( -4 + T \)
$17$ \( 24 + T \)
$19$ \( 44 + T \)
$23$ \( -72 + T \)
$29$ \( 38 + T \)
$31$ \( 184 + T \)
$37$ \( 30 + T \)
$41$ \( -216 + T \)
$43$ \( 164 + T \)
$47$ \( 520 + T \)
$53$ \( 146 + T \)
$59$ \( 460 + T \)
$61$ \( 628 + T \)
$67$ \( -556 + T \)
$71$ \( -592 + T \)
$73$ \( 1024 + T \)
$79$ \( 104 + T \)
$83$ \( -324 + T \)
$89$ \( 896 + T \)
$97$ \( -920 + T \)
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