Properties

Label 784.6.a.g
Level $784$
Weight $6$
Character orbit 784.a
Self dual yes
Analytic conductor $125.741$
Analytic rank $1$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 243 q^{9} + 76 q^{11} + 4952 q^{23} - 3125 q^{25} + 7282 q^{29} - 8886 q^{37} - 11748 q^{43} + 24550 q^{53} - 69364 q^{67} + 2224 q^{71} - 80168 q^{79} + 59049 q^{81} - 18468 q^{99}+O(q^{100}) \) 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 0 0 0 0 0 0 −243.000 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 784.6.a.g 1
4.b odd 2 1 49.6.a.b 1
7.b odd 2 1 CM 784.6.a.g 1
12.b even 2 1 441.6.a.a 1
28.d even 2 1 49.6.a.b 1
28.f even 6 2 49.6.c.a 2
28.g odd 6 2 49.6.c.a 2
84.h odd 2 1 441.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.b 1 4.b odd 2 1
49.6.a.b 1 28.d even 2 1
49.6.c.a 2 28.f even 6 2
49.6.c.a 2 28.g odd 6 2
441.6.a.a 1 12.b even 2 1
441.6.a.a 1 84.h odd 2 1
784.6.a.g 1 1.a even 1 1 trivial
784.6.a.g 1 7.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 76 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 4952 \) Copy content Toggle raw display
$29$ \( T - 7282 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 8886 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 11748 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 24550 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 69364 \) Copy content Toggle raw display
$71$ \( T - 2224 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 80168 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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