Properties

Label 784.6.a.l
Level $784$
Weight $6$
Character orbit 784.a
Self dual yes
Analytic conductor $125.741$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
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

\(f(q)\) \(=\) \( q + 20 q^{3} + 74 q^{5} + 157 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 20 q^{3} + 74 q^{5} + 157 q^{9} - 124 q^{11} - 478 q^{13} + 1480 q^{15} + 1198 q^{17} + 3044 q^{19} - 184 q^{23} + 2351 q^{25} - 1720 q^{27} - 3282 q^{29} - 5728 q^{31} - 2480 q^{33} + 10326 q^{37} - 9560 q^{39} + 8886 q^{41} + 9188 q^{43} + 11618 q^{45} + 23664 q^{47} + 23960 q^{51} + 11686 q^{53} - 9176 q^{55} + 60880 q^{57} + 16876 q^{59} + 18482 q^{61} - 35372 q^{65} + 15532 q^{67} - 3680 q^{69} + 31960 q^{71} + 4886 q^{73} + 47020 q^{75} - 44560 q^{79} - 72551 q^{81} + 67364 q^{83} + 88652 q^{85} - 65640 q^{87} - 71994 q^{89} - 114560 q^{93} + 225256 q^{95} - 48866 q^{97} - 19468 q^{99}+O(q^{100}) \) 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 20.0000 0 74.0000 0 0 0 157.000 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 784.6.a.l 1
4.b odd 2 1 392.6.a.b 1
7.b odd 2 1 16.6.a.a 1
21.c even 2 1 144.6.a.k 1
28.d even 2 1 8.6.a.a 1
28.f even 6 2 392.6.i.b 2
28.g odd 6 2 392.6.i.e 2
35.c odd 2 1 400.6.a.l 1
35.f even 4 2 400.6.c.d 2
56.e even 2 1 64.6.a.a 1
56.h odd 2 1 64.6.a.g 1
84.h odd 2 1 72.6.a.f 1
112.j even 4 2 256.6.b.f 2
112.l odd 4 2 256.6.b.d 2
140.c even 2 1 200.6.a.a 1
140.j odd 4 2 200.6.c.a 2
168.e odd 2 1 576.6.a.g 1
168.i even 2 1 576.6.a.h 1
308.g odd 2 1 968.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 28.d even 2 1
16.6.a.a 1 7.b odd 2 1
64.6.a.a 1 56.e even 2 1
64.6.a.g 1 56.h odd 2 1
72.6.a.f 1 84.h odd 2 1
144.6.a.k 1 21.c even 2 1
200.6.a.a 1 140.c even 2 1
200.6.c.a 2 140.j odd 4 2
256.6.b.d 2 112.l odd 4 2
256.6.b.f 2 112.j even 4 2
392.6.a.b 1 4.b odd 2 1
392.6.i.b 2 28.f even 6 2
392.6.i.e 2 28.g odd 6 2
400.6.a.l 1 35.c odd 2 1
400.6.c.d 2 35.f even 4 2
576.6.a.g 1 168.e odd 2 1
576.6.a.h 1 168.i even 2 1
784.6.a.l 1 1.a even 1 1 trivial
968.6.a.a 1 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 20 \) 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
$3$ \( T - 20 \) Copy content Toggle raw display
$5$ \( T - 74 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 124 \) Copy content Toggle raw display
$13$ \( T + 478 \) Copy content Toggle raw display
$17$ \( T - 1198 \) Copy content Toggle raw display
$19$ \( T - 3044 \) Copy content Toggle raw display
$23$ \( T + 184 \) Copy content Toggle raw display
$29$ \( T + 3282 \) Copy content Toggle raw display
$31$ \( T + 5728 \) Copy content Toggle raw display
$37$ \( T - 10326 \) Copy content Toggle raw display
$41$ \( T - 8886 \) Copy content Toggle raw display
$43$ \( T - 9188 \) Copy content Toggle raw display
$47$ \( T - 23664 \) Copy content Toggle raw display
$53$ \( T - 11686 \) Copy content Toggle raw display
$59$ \( T - 16876 \) Copy content Toggle raw display
$61$ \( T - 18482 \) Copy content Toggle raw display
$67$ \( T - 15532 \) Copy content Toggle raw display
$71$ \( T - 31960 \) Copy content Toggle raw display
$73$ \( T - 4886 \) Copy content Toggle raw display
$79$ \( T + 44560 \) Copy content Toggle raw display
$83$ \( T - 67364 \) Copy content Toggle raw display
$89$ \( T + 71994 \) Copy content Toggle raw display
$97$ \( T + 48866 \) Copy content Toggle raw display
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