Properties

Label 588.4.a.b
Level $588$
Weight $4$
Character orbit 588.a
Self dual yes
Analytic conductor $34.693$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - 3 q^{3} + 4 q^{5} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 4 q^{5} + 9 q^{9} - 20 q^{11} - 4 q^{13} - 12 q^{15} + 24 q^{17} + 44 q^{19} + 72 q^{23} - 109 q^{25} - 27 q^{27} - 38 q^{29} + 184 q^{31} + 60 q^{33} - 30 q^{37} + 12 q^{39} - 216 q^{41} - 164 q^{43} + 36 q^{45} + 520 q^{47} - 72 q^{51} - 146 q^{53} - 80 q^{55} - 132 q^{57} + 460 q^{59} + 628 q^{61} - 16 q^{65} + 556 q^{67} - 216 q^{69} + 592 q^{71} + 1024 q^{73} + 327 q^{75} - 104 q^{79} + 81 q^{81} - 324 q^{83} + 96 q^{85} + 114 q^{87} + 896 q^{89} - 552 q^{93} + 176 q^{95} - 920 q^{97} - 180 q^{99}+O(q^{100}) \) 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
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.b 1
3.b odd 2 1 1764.4.a.d 1
4.b odd 2 1 2352.4.a.be 1
7.b odd 2 1 588.4.a.e yes 1
7.c even 3 2 588.4.i.g 2
7.d odd 6 2 588.4.i.b 2
21.c even 2 1 1764.4.a.i 1
21.g even 6 2 1764.4.k.g 2
21.h odd 6 2 1764.4.k.j 2
28.d even 2 1 2352.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.b 1 1.a even 1 1 trivial
588.4.a.e yes 1 7.b odd 2 1
588.4.i.b 2 7.d odd 6 2
588.4.i.g 2 7.c even 3 2
1764.4.a.d 1 3.b odd 2 1
1764.4.a.i 1 21.c even 2 1
1764.4.k.g 2 21.g even 6 2
1764.4.k.j 2 21.h odd 6 2
2352.4.a.g 1 28.d even 2 1
2352.4.a.be 1 4.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

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