Properties

Label 784.4.a.b
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 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 7q^{3} + 7q^{5} + 22q^{9} + O(q^{10}) \) \( q - 7q^{3} + 7q^{5} + 22q^{9} + 5q^{11} - 14q^{13} - 49q^{15} - 21q^{17} - 49q^{19} + 159q^{23} - 76q^{25} + 35q^{27} + 58q^{29} - 147q^{31} - 35q^{33} + 219q^{37} + 98q^{39} + 350q^{41} + 124q^{43} + 154q^{45} - 525q^{47} + 147q^{51} + 303q^{53} + 35q^{55} + 343q^{57} + 105q^{59} - 413q^{61} - 98q^{65} - 415q^{67} - 1113q^{69} + 432q^{71} - 1113q^{73} + 532q^{75} + 103q^{79} - 839q^{81} - 1092q^{83} - 147q^{85} - 406q^{87} - 329q^{89} + 1029q^{93} - 343q^{95} - 882q^{97} + 110q^{99} + O(q^{100}) \)

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 −7.00000 0 7.00000 0 0 0 22.0000 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.b 1
4.b odd 2 1 49.4.a.d 1
7.b odd 2 1 784.4.a.r 1
7.c even 3 2 112.4.i.c 2
12.b even 2 1 441.4.a.d 1
20.d odd 2 1 1225.4.a.c 1
28.d even 2 1 49.4.a.c 1
28.f even 6 2 49.4.c.a 2
28.g odd 6 2 7.4.c.a 2
56.k odd 6 2 448.4.i.f 2
56.p even 6 2 448.4.i.a 2
84.h odd 2 1 441.4.a.e 1
84.j odd 6 2 441.4.e.k 2
84.n even 6 2 63.4.e.b 2
140.c even 2 1 1225.4.a.d 1
140.p odd 6 2 175.4.e.a 2
140.w even 12 4 175.4.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 28.g odd 6 2
49.4.a.c 1 28.d even 2 1
49.4.a.d 1 4.b odd 2 1
49.4.c.a 2 28.f even 6 2
63.4.e.b 2 84.n even 6 2
112.4.i.c 2 7.c even 3 2
175.4.e.a 2 140.p odd 6 2
175.4.k.a 4 140.w even 12 4
441.4.a.d 1 12.b even 2 1
441.4.a.e 1 84.h odd 2 1
441.4.e.k 2 84.j odd 6 2
448.4.i.a 2 56.p even 6 2
448.4.i.f 2 56.k odd 6 2
784.4.a.b 1 1.a even 1 1 trivial
784.4.a.r 1 7.b odd 2 1
1225.4.a.c 1 20.d odd 2 1
1225.4.a.d 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3} + 7 \)
\( T_{5} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 7 + T \)
$5$ \( -7 + T \)
$7$ \( T \)
$11$ \( -5 + T \)
$13$ \( 14 + T \)
$17$ \( 21 + T \)
$19$ \( 49 + T \)
$23$ \( -159 + T \)
$29$ \( -58 + T \)
$31$ \( 147 + T \)
$37$ \( -219 + T \)
$41$ \( -350 + T \)
$43$ \( -124 + T \)
$47$ \( 525 + T \)
$53$ \( -303 + T \)
$59$ \( -105 + T \)
$61$ \( 413 + T \)
$67$ \( 415 + T \)
$71$ \( -432 + T \)
$73$ \( 1113 + T \)
$79$ \( -103 + T \)
$83$ \( 1092 + T \)
$89$ \( 329 + T \)
$97$ \( 882 + T \)
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