Properties

Label 392.8.a.d
Level $392$
Weight $8$
Character orbit 392.a
Self dual yes
Analytic conductor $122.455$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(122.454929990\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 84q^{3} + 82q^{5} + 4869q^{9} + O(q^{10}) \) \( q + 84q^{3} + 82q^{5} + 4869q^{9} - 2524q^{11} + 10778q^{13} + 6888q^{15} + 11150q^{17} - 4124q^{19} + 81704q^{23} - 71401q^{25} + 225288q^{27} + 99798q^{29} + 40480q^{31} - 212016q^{33} - 419442q^{37} + 905352q^{39} - 141402q^{41} - 690428q^{43} + 399258q^{45} + 682032q^{47} + 936600q^{51} + 1813118q^{53} - 206968q^{55} - 346416q^{57} + 966028q^{59} - 1887670q^{61} + 883796q^{65} + 2965868q^{67} + 6863136q^{69} - 2548232q^{71} + 1680326q^{73} - 5997684q^{75} + 4038064q^{79} + 8275689q^{81} + 5385764q^{83} + 914300q^{85} + 8383032q^{87} + 6473046q^{89} + 3400320q^{93} - 338168q^{95} + 6065758q^{97} - 12289356q^{99} + O(q^{100}) \)

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 84.0000 0 82.0000 0 0 0 4869.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 392.8.a.d 1
7.b odd 2 1 8.8.a.a 1
21.c even 2 1 72.8.a.d 1
28.d even 2 1 16.8.a.c 1
35.c odd 2 1 200.8.a.i 1
35.f even 4 2 200.8.c.a 2
56.e even 2 1 64.8.a.a 1
56.h odd 2 1 64.8.a.g 1
84.h odd 2 1 144.8.a.g 1
112.j even 4 2 256.8.b.c 2
112.l odd 4 2 256.8.b.e 2
140.c even 2 1 400.8.a.b 1
140.j odd 4 2 400.8.c.b 2
168.e odd 2 1 576.8.a.k 1
168.i even 2 1 576.8.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 7.b odd 2 1
16.8.a.c 1 28.d even 2 1
64.8.a.a 1 56.e even 2 1
64.8.a.g 1 56.h odd 2 1
72.8.a.d 1 21.c even 2 1
144.8.a.g 1 84.h odd 2 1
200.8.a.i 1 35.c odd 2 1
200.8.c.a 2 35.f even 4 2
256.8.b.c 2 112.j even 4 2
256.8.b.e 2 112.l odd 4 2
392.8.a.d 1 1.a even 1 1 trivial
400.8.a.b 1 140.c even 2 1
400.8.c.b 2 140.j odd 4 2
576.8.a.j 1 168.i even 2 1
576.8.a.k 1 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 84 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(392))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -84 + T \)
$5$ \( -82 + T \)
$7$ \( T \)
$11$ \( 2524 + T \)
$13$ \( -10778 + T \)
$17$ \( -11150 + T \)
$19$ \( 4124 + T \)
$23$ \( -81704 + T \)
$29$ \( -99798 + T \)
$31$ \( -40480 + T \)
$37$ \( 419442 + T \)
$41$ \( 141402 + T \)
$43$ \( 690428 + T \)
$47$ \( -682032 + T \)
$53$ \( -1813118 + T \)
$59$ \( -966028 + T \)
$61$ \( 1887670 + T \)
$67$ \( -2965868 + T \)
$71$ \( 2548232 + T \)
$73$ \( -1680326 + T \)
$79$ \( -4038064 + T \)
$83$ \( -5385764 + T \)
$89$ \( -6473046 + T \)
$97$ \( -6065758 + T \)
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