Properties

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

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} - 16q^{5} - 23q^{9} + O(q^{10}) \) \( q - 2q^{3} - 16q^{5} - 23q^{9} + 8q^{11} - 28q^{13} + 32q^{15} - 54q^{17} - 110q^{19} - 48q^{23} + 131q^{25} + 100q^{27} - 110q^{29} + 12q^{31} - 16q^{33} - 246q^{37} + 56q^{39} - 182q^{41} - 128q^{43} + 368q^{45} + 324q^{47} + 108q^{51} - 162q^{53} - 128q^{55} + 220q^{57} + 810q^{59} + 488q^{61} + 448q^{65} - 244q^{67} + 96q^{69} + 768q^{71} + 702q^{73} - 262q^{75} - 440q^{79} + 421q^{81} - 1302q^{83} + 864q^{85} + 220q^{87} - 730q^{89} - 24q^{93} + 1760q^{95} - 294q^{97} - 184q^{99} + O(q^{100}) \)

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 −2.00000 0 −16.0000 0 0 0 −23.0000 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.g 1
4.b odd 2 1 49.4.a.b 1
7.b odd 2 1 112.4.a.f 1
12.b even 2 1 441.4.a.i 1
20.d odd 2 1 1225.4.a.j 1
21.c even 2 1 1008.4.a.c 1
28.d even 2 1 7.4.a.a 1
28.f even 6 2 49.4.c.c 2
28.g odd 6 2 49.4.c.b 2
56.e even 2 1 448.4.a.i 1
56.h odd 2 1 448.4.a.e 1
84.h odd 2 1 63.4.a.b 1
84.j odd 6 2 441.4.e.h 2
84.n even 6 2 441.4.e.e 2
140.c even 2 1 175.4.a.b 1
140.j odd 4 2 175.4.b.b 2
308.g odd 2 1 847.4.a.b 1
364.h even 2 1 1183.4.a.b 1
420.o odd 2 1 1575.4.a.e 1
476.e even 2 1 2023.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 28.d even 2 1
49.4.a.b 1 4.b odd 2 1
49.4.c.b 2 28.g odd 6 2
49.4.c.c 2 28.f even 6 2
63.4.a.b 1 84.h odd 2 1
112.4.a.f 1 7.b odd 2 1
175.4.a.b 1 140.c even 2 1
175.4.b.b 2 140.j odd 4 2
441.4.a.i 1 12.b even 2 1
441.4.e.e 2 84.n even 6 2
441.4.e.h 2 84.j odd 6 2
448.4.a.e 1 56.h odd 2 1
448.4.a.i 1 56.e even 2 1
784.4.a.g 1 1.a even 1 1 trivial
847.4.a.b 1 308.g odd 2 1
1008.4.a.c 1 21.c even 2 1
1183.4.a.b 1 364.h even 2 1
1225.4.a.j 1 20.d odd 2 1
1575.4.a.e 1 420.o odd 2 1
2023.4.a.a 1 476.e even 2 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{5} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( 16 + T \)
$7$ \( T \)
$11$ \( -8 + T \)
$13$ \( 28 + T \)
$17$ \( 54 + T \)
$19$ \( 110 + T \)
$23$ \( 48 + T \)
$29$ \( 110 + T \)
$31$ \( -12 + T \)
$37$ \( 246 + T \)
$41$ \( 182 + T \)
$43$ \( 128 + T \)
$47$ \( -324 + T \)
$53$ \( 162 + T \)
$59$ \( -810 + T \)
$61$ \( -488 + T \)
$67$ \( 244 + T \)
$71$ \( -768 + T \)
$73$ \( -702 + T \)
$79$ \( 440 + T \)
$83$ \( 1302 + T \)
$89$ \( 730 + T \)
$97$ \( 294 + T \)
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