Properties

Label 784.4.a.k
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 27q^{9} + O(q^{10}) \) \( q - 27q^{9} + 68q^{11} + 40q^{23} - 125q^{25} - 166q^{29} + 450q^{37} + 180q^{43} + 590q^{53} + 740q^{67} - 688q^{71} + 1384q^{79} + 729q^{81} - 1836q^{99} + O(q^{100}) \)

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 0 0 0 0 0 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.k 1
4.b odd 2 1 49.4.a.a 1
7.b odd 2 1 CM 784.4.a.k 1
12.b even 2 1 441.4.a.m 1
20.d odd 2 1 1225.4.a.l 1
28.d even 2 1 49.4.a.a 1
28.f even 6 2 49.4.c.d 2
28.g odd 6 2 49.4.c.d 2
84.h odd 2 1 441.4.a.m 1
84.j odd 6 2 441.4.e.a 2
84.n even 6 2 441.4.e.a 2
140.c even 2 1 1225.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 4.b odd 2 1
49.4.a.a 1 28.d even 2 1
49.4.c.d 2 28.f even 6 2
49.4.c.d 2 28.g odd 6 2
441.4.a.m 1 12.b even 2 1
441.4.a.m 1 84.h odd 2 1
441.4.e.a 2 84.j odd 6 2
441.4.e.a 2 84.n even 6 2
784.4.a.k 1 1.a even 1 1 trivial
784.4.a.k 1 7.b odd 2 1 CM
1225.4.a.l 1 20.d odd 2 1
1225.4.a.l 1 140.c even 2 1

Hecke kernels

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

\( T_{3} \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -68 + T \)
$13$ \( T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( -40 + T \)
$29$ \( 166 + T \)
$31$ \( T \)
$37$ \( -450 + T \)
$41$ \( T \)
$43$ \( -180 + T \)
$47$ \( T \)
$53$ \( -590 + T \)
$59$ \( T \)
$61$ \( T \)
$67$ \( -740 + T \)
$71$ \( 688 + T \)
$73$ \( T \)
$79$ \( -1384 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( T \)
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