Properties

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

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 7q^{5} - 26q^{9} + O(q^{10}) \) \( q + q^{3} + 7q^{5} - 26q^{9} - 35q^{11} + 66q^{13} + 7q^{15} + 59q^{17} - 137q^{19} + 7q^{23} - 76q^{25} - 53q^{27} + 106q^{29} - 75q^{31} - 35q^{33} + 11q^{37} + 66q^{39} - 498q^{41} - 260q^{43} - 182q^{45} + 171q^{47} + 59q^{51} - 417q^{53} - 245q^{55} - 137q^{57} + 17q^{59} + 51q^{61} + 462q^{65} - 439q^{67} + 7q^{69} + 784q^{71} + 295q^{73} - 76q^{75} + 495q^{79} + 649q^{81} - 932q^{83} + 413q^{85} + 106q^{87} - 873q^{89} - 75q^{93} - 959q^{95} - 290q^{97} + 910q^{99} + O(q^{100}) \)

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 1.00000 0 7.00000 0 0 0 −26.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.l 1
4.b odd 2 1 98.4.a.b 1
7.b odd 2 1 784.4.a.j 1
7.c even 3 2 112.4.i.b 2
12.b even 2 1 882.4.a.k 1
20.d odd 2 1 2450.4.a.bh 1
28.d even 2 1 98.4.a.c 1
28.f even 6 2 98.4.c.e 2
28.g odd 6 2 14.4.c.b 2
56.k odd 6 2 448.4.i.c 2
56.p even 6 2 448.4.i.d 2
84.h odd 2 1 882.4.a.p 1
84.j odd 6 2 882.4.g.d 2
84.n even 6 2 126.4.g.c 2
140.c even 2 1 2450.4.a.bf 1
140.p odd 6 2 350.4.e.b 2
140.w even 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.g odd 6 2
98.4.a.b 1 4.b odd 2 1
98.4.a.c 1 28.d even 2 1
98.4.c.e 2 28.f even 6 2
112.4.i.b 2 7.c even 3 2
126.4.g.c 2 84.n even 6 2
350.4.e.b 2 140.p odd 6 2
350.4.j.d 4 140.w even 12 4
448.4.i.c 2 56.k odd 6 2
448.4.i.d 2 56.p even 6 2
784.4.a.j 1 7.b odd 2 1
784.4.a.l 1 1.a even 1 1 trivial
882.4.a.k 1 12.b even 2 1
882.4.a.p 1 84.h odd 2 1
882.4.g.d 2 84.j odd 6 2
2450.4.a.bf 1 140.c even 2 1
2450.4.a.bh 1 20.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{5} - 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + 27 T^{2} \)
$5$ \( 1 - 7 T + 125 T^{2} \)
$7$ 1
$11$ \( 1 + 35 T + 1331 T^{2} \)
$13$ \( 1 - 66 T + 2197 T^{2} \)
$17$ \( 1 - 59 T + 4913 T^{2} \)
$19$ \( 1 + 137 T + 6859 T^{2} \)
$23$ \( 1 - 7 T + 12167 T^{2} \)
$29$ \( 1 - 106 T + 24389 T^{2} \)
$31$ \( 1 + 75 T + 29791 T^{2} \)
$37$ \( 1 - 11 T + 50653 T^{2} \)
$41$ \( 1 + 498 T + 68921 T^{2} \)
$43$ \( 1 + 260 T + 79507 T^{2} \)
$47$ \( 1 - 171 T + 103823 T^{2} \)
$53$ \( 1 + 417 T + 148877 T^{2} \)
$59$ \( 1 - 17 T + 205379 T^{2} \)
$61$ \( 1 - 51 T + 226981 T^{2} \)
$67$ \( 1 + 439 T + 300763 T^{2} \)
$71$ \( 1 - 784 T + 357911 T^{2} \)
$73$ \( 1 - 295 T + 389017 T^{2} \)
$79$ \( 1 - 495 T + 493039 T^{2} \)
$83$ \( 1 + 932 T + 571787 T^{2} \)
$89$ \( 1 + 873 T + 704969 T^{2} \)
$97$ \( 1 + 290 T + 912673 T^{2} \)
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