Properties

Label 784.4.a.p
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q + 5q^{3} - 9q^{5} - 2q^{9} + O(q^{10}) \) \( q + 5q^{3} - 9q^{5} - 2q^{9} + 57q^{11} - 70q^{13} - 45q^{15} + 51q^{17} - 5q^{19} - 69q^{23} - 44q^{25} - 145q^{27} + 114q^{29} - 23q^{31} + 285q^{33} - 253q^{37} - 350q^{39} - 42q^{41} + 124q^{43} + 18q^{45} - 201q^{47} + 255q^{51} - 393q^{53} - 513q^{55} - 25q^{57} - 219q^{59} - 709q^{61} + 630q^{65} - 419q^{67} - 345q^{69} + 96q^{71} - 313q^{73} - 220q^{75} - 461q^{79} - 671q^{81} + 588q^{83} - 459q^{85} + 570q^{87} - 1017q^{89} - 115q^{93} + 45q^{95} - 1834q^{97} - 114q^{99} + O(q^{100}) \)

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 5.00000 0 −9.00000 0 0 0 −2.00000 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.p 1
4.b odd 2 1 98.4.a.d 1
7.b odd 2 1 784.4.a.c 1
7.c even 3 2 112.4.i.a 2
12.b even 2 1 882.4.a.f 1
20.d odd 2 1 2450.4.a.q 1
28.d even 2 1 98.4.a.f 1
28.f even 6 2 98.4.c.a 2
28.g odd 6 2 14.4.c.a 2
56.k odd 6 2 448.4.i.b 2
56.p even 6 2 448.4.i.e 2
84.h odd 2 1 882.4.a.c 1
84.j odd 6 2 882.4.g.u 2
84.n even 6 2 126.4.g.d 2
140.c even 2 1 2450.4.a.d 1
140.p odd 6 2 350.4.e.e 2
140.w even 12 4 350.4.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 28.g odd 6 2
98.4.a.d 1 4.b odd 2 1
98.4.a.f 1 28.d even 2 1
98.4.c.a 2 28.f even 6 2
112.4.i.a 2 7.c even 3 2
126.4.g.d 2 84.n even 6 2
350.4.e.e 2 140.p odd 6 2
350.4.j.b 4 140.w even 12 4
448.4.i.b 2 56.k odd 6 2
448.4.i.e 2 56.p even 6 2
784.4.a.c 1 7.b odd 2 1
784.4.a.p 1 1.a even 1 1 trivial
882.4.a.c 1 84.h odd 2 1
882.4.a.f 1 12.b even 2 1
882.4.g.u 2 84.j odd 6 2
2450.4.a.d 1 140.c even 2 1
2450.4.a.q 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3} - 5 \)
\( T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -5 + T \)
$5$ \( 9 + T \)
$7$ \( T \)
$11$ \( -57 + T \)
$13$ \( 70 + T \)
$17$ \( -51 + T \)
$19$ \( 5 + T \)
$23$ \( 69 + T \)
$29$ \( -114 + T \)
$31$ \( 23 + T \)
$37$ \( 253 + T \)
$41$ \( 42 + T \)
$43$ \( -124 + T \)
$47$ \( 201 + T \)
$53$ \( 393 + T \)
$59$ \( 219 + T \)
$61$ \( 709 + T \)
$67$ \( 419 + T \)
$71$ \( -96 + T \)
$73$ \( 313 + T \)
$79$ \( 461 + T \)
$83$ \( -588 + T \)
$89$ \( 1017 + T \)
$97$ \( 1834 + T \)
show more
show less