Properties

Label 784.4.a.s
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

\(f(q)\) \(=\) \( q + 8q^{3} + 14q^{5} + 37q^{9} + O(q^{10}) \) \( q + 8q^{3} + 14q^{5} + 37q^{9} + 28q^{11} - 18q^{13} + 112q^{15} - 74q^{17} + 80q^{19} + 112q^{23} + 71q^{25} + 80q^{27} + 190q^{29} + 72q^{31} + 224q^{33} - 346q^{37} - 144q^{39} - 162q^{41} + 412q^{43} + 518q^{45} + 24q^{47} - 592q^{51} + 318q^{53} + 392q^{55} + 640q^{57} - 200q^{59} + 198q^{61} - 252q^{65} + 716q^{67} + 896q^{69} - 392q^{71} - 538q^{73} + 568q^{75} - 240q^{79} - 359q^{81} - 1072q^{83} - 1036q^{85} + 1520q^{87} - 810q^{89} + 576q^{93} + 1120q^{95} - 1354q^{97} + 1036q^{99} + O(q^{100}) \)

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 8.00000 0 14.0000 0 0 0 37.0000 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.s 1
4.b odd 2 1 98.4.a.a 1
7.b odd 2 1 112.4.a.a 1
12.b even 2 1 882.4.a.i 1
20.d odd 2 1 2450.4.a.bo 1
21.c even 2 1 1008.4.a.s 1
28.d even 2 1 14.4.a.a 1
28.f even 6 2 98.4.c.d 2
28.g odd 6 2 98.4.c.f 2
56.e even 2 1 448.4.a.b 1
56.h odd 2 1 448.4.a.o 1
84.h odd 2 1 126.4.a.h 1
84.j odd 6 2 882.4.g.b 2
84.n even 6 2 882.4.g.k 2
140.c even 2 1 350.4.a.l 1
140.j odd 4 2 350.4.c.b 2
308.g odd 2 1 1694.4.a.g 1
364.h even 2 1 2366.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 28.d even 2 1
98.4.a.a 1 4.b odd 2 1
98.4.c.d 2 28.f even 6 2
98.4.c.f 2 28.g odd 6 2
112.4.a.a 1 7.b odd 2 1
126.4.a.h 1 84.h odd 2 1
350.4.a.l 1 140.c even 2 1
350.4.c.b 2 140.j odd 4 2
448.4.a.b 1 56.e even 2 1
448.4.a.o 1 56.h odd 2 1
784.4.a.s 1 1.a even 1 1 trivial
882.4.a.i 1 12.b even 2 1
882.4.g.b 2 84.j odd 6 2
882.4.g.k 2 84.n even 6 2
1008.4.a.s 1 21.c even 2 1
1694.4.a.g 1 308.g odd 2 1
2366.4.a.h 1 364.h even 2 1
2450.4.a.bo 1 20.d odd 2 1

Hecke kernels

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

\( T_{3} - 8 \)
\( T_{5} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -8 + T \)
$5$ \( -14 + T \)
$7$ \( T \)
$11$ \( -28 + T \)
$13$ \( 18 + T \)
$17$ \( 74 + T \)
$19$ \( -80 + T \)
$23$ \( -112 + T \)
$29$ \( -190 + T \)
$31$ \( -72 + T \)
$37$ \( 346 + T \)
$41$ \( 162 + T \)
$43$ \( -412 + T \)
$47$ \( -24 + T \)
$53$ \( -318 + T \)
$59$ \( 200 + T \)
$61$ \( -198 + T \)
$67$ \( -716 + T \)
$71$ \( 392 + T \)
$73$ \( 538 + T \)
$79$ \( 240 + T \)
$83$ \( 1072 + T \)
$89$ \( 810 + T \)
$97$ \( 1354 + T \)
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