Properties

Label 784.4.a.d
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.2574974445\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 392)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{3} - 12q^{5} - 11q^{9} + O(q^{10}) \) \( q - 4q^{3} - 12q^{5} - 11q^{9} - 12q^{11} + 76q^{13} + 48q^{15} - 8q^{17} + 100q^{19} + 56q^{23} + 19q^{25} + 152q^{27} - 166q^{29} + 232q^{31} + 48q^{33} - 414q^{37} - 304q^{39} + 72q^{41} + 452q^{43} + 132q^{45} - 424q^{47} + 32q^{51} - 18q^{53} + 144q^{55} - 400q^{57} - 444q^{59} - 284q^{61} - 912q^{65} - 524q^{67} - 224q^{69} + 1008q^{71} + 896q^{73} - 76q^{75} + 40q^{79} - 311q^{81} - 1388q^{83} + 96q^{85} + 664q^{87} + 448q^{89} - 928q^{93} - 1200q^{95} - 824q^{97} + 132q^{99} + O(q^{100}) \)

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 −4.00000 0 −12.0000 0 0 0 −11.0000 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 784.4.a.d 1
4.b odd 2 1 392.4.a.d yes 1
7.b odd 2 1 784.4.a.o 1
28.d even 2 1 392.4.a.b 1
28.f even 6 2 392.4.i.f 2
28.g odd 6 2 392.4.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.b 1 28.d even 2 1
392.4.a.d yes 1 4.b odd 2 1
392.4.i.c 2 28.g odd 6 2
392.4.i.f 2 28.f even 6 2
784.4.a.d 1 1.a even 1 1 trivial
784.4.a.o 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3} + 4 \)
\( T_{5} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 4 + T \)
$5$ \( 12 + T \)
$7$ \( T \)
$11$ \( 12 + T \)
$13$ \( -76 + T \)
$17$ \( 8 + T \)
$19$ \( -100 + T \)
$23$ \( -56 + T \)
$29$ \( 166 + T \)
$31$ \( -232 + T \)
$37$ \( 414 + T \)
$41$ \( -72 + T \)
$43$ \( -452 + T \)
$47$ \( 424 + T \)
$53$ \( 18 + T \)
$59$ \( 444 + T \)
$61$ \( 284 + T \)
$67$ \( 524 + T \)
$71$ \( -1008 + T \)
$73$ \( -896 + T \)
$79$ \( -40 + T \)
$83$ \( 1388 + T \)
$89$ \( -448 + T \)
$97$ \( 824 + T \)
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