Properties

Label 784.4.a.q
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 about

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 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 6q^{3} - 8q^{5} + 9q^{9} + O(q^{10}) \) \( q + 6q^{3} - 8q^{5} + 9q^{9} - 56q^{11} + 28q^{13} - 48q^{15} + 90q^{17} + 74q^{19} + 96q^{23} - 61q^{25} - 108q^{27} - 222q^{29} - 100q^{31} - 336q^{33} + 58q^{37} + 168q^{39} - 422q^{41} - 512q^{43} - 72q^{45} + 148q^{47} + 540q^{51} - 642q^{53} + 448q^{55} + 444q^{57} - 318q^{59} - 720q^{61} - 224q^{65} + 412q^{67} + 576q^{69} - 448q^{71} - 994q^{73} - 366q^{75} + 296q^{79} - 891q^{81} + 386q^{83} - 720q^{85} - 1332q^{87} + 6q^{89} - 600q^{93} - 592q^{95} + 138q^{97} - 504q^{99} + O(q^{100}) \)

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 6.00000 0 −8.00000 0 0 0 9.00000 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.q 1
4.b odd 2 1 392.4.a.a 1
7.b odd 2 1 112.4.a.b 1
21.c even 2 1 1008.4.a.e 1
28.d even 2 1 56.4.a.b 1
28.f even 6 2 392.4.i.a 2
28.g odd 6 2 392.4.i.h 2
56.e even 2 1 448.4.a.c 1
56.h odd 2 1 448.4.a.n 1
84.h odd 2 1 504.4.a.a 1
140.c even 2 1 1400.4.a.b 1
140.j odd 4 2 1400.4.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.b 1 28.d even 2 1
112.4.a.b 1 7.b odd 2 1
392.4.a.a 1 4.b odd 2 1
392.4.i.a 2 28.f even 6 2
392.4.i.h 2 28.g odd 6 2
448.4.a.c 1 56.e even 2 1
448.4.a.n 1 56.h odd 2 1
504.4.a.a 1 84.h odd 2 1
784.4.a.q 1 1.a even 1 1 trivial
1008.4.a.e 1 21.c even 2 1
1400.4.a.b 1 140.c even 2 1
1400.4.g.b 2 140.j odd 4 2

Hecke kernels

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

\( T_{3} - 6 \)
\( T_{5} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -6 + T \)
$5$ \( 8 + T \)
$7$ \( T \)
$11$ \( 56 + T \)
$13$ \( -28 + T \)
$17$ \( -90 + T \)
$19$ \( -74 + T \)
$23$ \( -96 + T \)
$29$ \( 222 + T \)
$31$ \( 100 + T \)
$37$ \( -58 + T \)
$41$ \( 422 + T \)
$43$ \( 512 + T \)
$47$ \( -148 + T \)
$53$ \( 642 + T \)
$59$ \( 318 + T \)
$61$ \( 720 + T \)
$67$ \( -412 + T \)
$71$ \( 448 + T \)
$73$ \( 994 + T \)
$79$ \( -296 + T \)
$83$ \( -386 + T \)
$89$ \( -6 + T \)
$97$ \( -138 + T \)
show more
show less