Properties

Label 784.4.a.j
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 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 7 q^{5} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 7 q^{5} - 26 q^{9} - 35 q^{11} - 66 q^{13} + 7 q^{15} - 59 q^{17} + 137 q^{19} + 7 q^{23} - 76 q^{25} + 53 q^{27} + 106 q^{29} + 75 q^{31} + 35 q^{33} + 11 q^{37} + 66 q^{39} + 498 q^{41} - 260 q^{43} + 182 q^{45} - 171 q^{47} + 59 q^{51} - 417 q^{53} + 245 q^{55} - 137 q^{57} - 17 q^{59} - 51 q^{61} + 462 q^{65} - 439 q^{67} - 7 q^{69} + 784 q^{71} - 295 q^{73} + 76 q^{75} + 495 q^{79} + 649 q^{81} + 932 q^{83} + 413 q^{85} - 106 q^{87} + 873 q^{89} - 75 q^{93} - 959 q^{95} + 290 q^{97} + 910 q^{99}+O(q^{100}) \) 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 −1.00000 0 −7.00000 0 0 0 −26.0000 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.j 1
4.b odd 2 1 98.4.a.c 1
7.b odd 2 1 784.4.a.l 1
7.d odd 6 2 112.4.i.b 2
12.b even 2 1 882.4.a.p 1
20.d odd 2 1 2450.4.a.bf 1
28.d even 2 1 98.4.a.b 1
28.f even 6 2 14.4.c.b 2
28.g odd 6 2 98.4.c.e 2
56.j odd 6 2 448.4.i.d 2
56.m even 6 2 448.4.i.c 2
84.h odd 2 1 882.4.a.k 1
84.j odd 6 2 126.4.g.c 2
84.n even 6 2 882.4.g.d 2
140.c even 2 1 2450.4.a.bh 1
140.s even 6 2 350.4.e.b 2
140.x odd 12 4 350.4.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 28.f even 6 2
98.4.a.b 1 28.d even 2 1
98.4.a.c 1 4.b odd 2 1
98.4.c.e 2 28.g odd 6 2
112.4.i.b 2 7.d odd 6 2
126.4.g.c 2 84.j odd 6 2
350.4.e.b 2 140.s even 6 2
350.4.j.d 4 140.x odd 12 4
448.4.i.c 2 56.m even 6 2
448.4.i.d 2 56.j odd 6 2
784.4.a.j 1 1.a even 1 1 trivial
784.4.a.l 1 7.b odd 2 1
882.4.a.k 1 84.h odd 2 1
882.4.a.p 1 12.b even 2 1
882.4.g.d 2 84.n even 6 2
2450.4.a.bf 1 20.d odd 2 1
2450.4.a.bh 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 7 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 35 \) Copy content Toggle raw display
$13$ \( T + 66 \) Copy content Toggle raw display
$17$ \( T + 59 \) Copy content Toggle raw display
$19$ \( T - 137 \) Copy content Toggle raw display
$23$ \( T - 7 \) Copy content Toggle raw display
$29$ \( T - 106 \) Copy content Toggle raw display
$31$ \( T - 75 \) Copy content Toggle raw display
$37$ \( T - 11 \) Copy content Toggle raw display
$41$ \( T - 498 \) Copy content Toggle raw display
$43$ \( T + 260 \) Copy content Toggle raw display
$47$ \( T + 171 \) Copy content Toggle raw display
$53$ \( T + 417 \) Copy content Toggle raw display
$59$ \( T + 17 \) Copy content Toggle raw display
$61$ \( T + 51 \) Copy content Toggle raw display
$67$ \( T + 439 \) Copy content Toggle raw display
$71$ \( T - 784 \) Copy content Toggle raw display
$73$ \( T + 295 \) Copy content Toggle raw display
$79$ \( T - 495 \) Copy content Toggle raw display
$83$ \( T - 932 \) Copy content Toggle raw display
$89$ \( T - 873 \) Copy content Toggle raw display
$97$ \( T - 290 \) Copy content Toggle raw display
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