Properties

Label 784.4.a.n
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
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: \( k \) \(=\) \( 4 \)
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 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{3} - 6q^{5} - 11q^{9} + O(q^{10}) \) \( q + 4q^{3} - 6q^{5} - 11q^{9} + 12q^{11} + 82q^{13} - 24q^{15} + 30q^{17} + 68q^{19} - 216q^{23} - 89q^{25} - 152q^{27} + 246q^{29} - 112q^{31} + 48q^{33} + 110q^{37} + 328q^{39} + 246q^{41} + 172q^{43} + 66q^{45} + 192q^{47} + 120q^{51} + 558q^{53} - 72q^{55} + 272q^{57} + 540q^{59} - 110q^{61} - 492q^{65} - 140q^{67} - 864q^{69} + 840q^{71} + 550q^{73} - 356q^{75} + 208q^{79} - 311q^{81} + 516q^{83} - 180q^{85} + 984q^{87} + 1398q^{89} - 448q^{93} - 408q^{95} - 1586q^{97} - 132q^{99} + O(q^{100}) \)

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 −6.00000 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.n 1
4.b odd 2 1 196.4.a.b 1
7.b odd 2 1 112.4.a.c 1
12.b even 2 1 1764.4.a.k 1
21.c even 2 1 1008.4.a.f 1
28.d even 2 1 28.4.a.b 1
28.f even 6 2 196.4.e.c 2
28.g odd 6 2 196.4.e.d 2
56.e even 2 1 448.4.a.d 1
56.h odd 2 1 448.4.a.m 1
84.h odd 2 1 252.4.a.c 1
84.j odd 6 2 1764.4.k.k 2
84.n even 6 2 1764.4.k.e 2
140.c even 2 1 700.4.a.e 1
140.j odd 4 2 700.4.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 28.d even 2 1
112.4.a.c 1 7.b odd 2 1
196.4.a.b 1 4.b odd 2 1
196.4.e.c 2 28.f even 6 2
196.4.e.d 2 28.g odd 6 2
252.4.a.c 1 84.h odd 2 1
448.4.a.d 1 56.e even 2 1
448.4.a.m 1 56.h odd 2 1
700.4.a.e 1 140.c even 2 1
700.4.e.f 2 140.j odd 4 2
784.4.a.n 1 1.a even 1 1 trivial
1008.4.a.f 1 21.c even 2 1
1764.4.a.k 1 12.b even 2 1
1764.4.k.e 2 84.n even 6 2
1764.4.k.k 2 84.j odd 6 2

Hecke kernels

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

\( T_{3} - 4 \)
\( T_{5} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -4 + T \)
$5$ \( 6 + T \)
$7$ \( T \)
$11$ \( -12 + T \)
$13$ \( -82 + T \)
$17$ \( -30 + T \)
$19$ \( -68 + T \)
$23$ \( 216 + T \)
$29$ \( -246 + T \)
$31$ \( 112 + T \)
$37$ \( -110 + T \)
$41$ \( -246 + T \)
$43$ \( -172 + T \)
$47$ \( -192 + T \)
$53$ \( -558 + T \)
$59$ \( -540 + T \)
$61$ \( 110 + T \)
$67$ \( 140 + T \)
$71$ \( -840 + T \)
$73$ \( -550 + T \)
$79$ \( -208 + T \)
$83$ \( -516 + T \)
$89$ \( -1398 + T \)
$97$ \( 1586 + T \)
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