Properties

Label 784.4.a.v
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 392)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 10 \beta q^{5} -25 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 10 \beta q^{5} -25 q^{9} -54 q^{11} + 16 \beta q^{13} + 20 q^{15} + 23 \beta q^{17} + 45 \beta q^{19} -28 q^{23} + 75 q^{25} -52 \beta q^{27} + 282 q^{29} + 194 \beta q^{31} -54 \beta q^{33} + 146 q^{37} + 32 q^{39} + 241 \beta q^{41} -10 q^{43} -250 \beta q^{45} + 358 \beta q^{47} + 46 q^{51} + 598 q^{53} -540 \beta q^{55} + 90 q^{57} -407 \beta q^{59} + 330 \beta q^{61} + 320 q^{65} -916 q^{67} -28 \beta q^{69} -420 q^{71} -497 \beta q^{73} + 75 \beta q^{75} + 292 q^{79} + 571 q^{81} -815 \beta q^{83} + 460 q^{85} + 282 \beta q^{87} + 315 \beta q^{89} + 388 q^{93} + 900 q^{95} -417 \beta q^{97} + 1350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 50q^{9} + O(q^{10}) \) \( 2q - 50q^{9} - 108q^{11} + 40q^{15} - 56q^{23} + 150q^{25} + 564q^{29} + 292q^{37} + 64q^{39} - 20q^{43} + 92q^{51} + 1196q^{53} + 180q^{57} + 640q^{65} - 1832q^{67} - 840q^{71} + 584q^{79} + 1142q^{81} + 920q^{85} + 776q^{93} + 1800q^{95} + 2700q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.41421 0 −14.1421 0 0 0 −25.0000 0
1.2 0 1.41421 0 14.1421 0 0 0 −25.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.v 2
4.b odd 2 1 392.4.a.f 2
7.b odd 2 1 inner 784.4.a.v 2
28.d even 2 1 392.4.a.f 2
28.f even 6 2 392.4.i.k 4
28.g odd 6 2 392.4.i.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.f 2 4.b odd 2 1
392.4.a.f 2 28.d even 2 1
392.4.i.k 4 28.f even 6 2
392.4.i.k 4 28.g odd 6 2
784.4.a.v 2 1.a even 1 1 trivial
784.4.a.v 2 7.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} - 200 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( -200 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 54 + T )^{2} \)
$13$ \( -512 + T^{2} \)
$17$ \( -1058 + T^{2} \)
$19$ \( -4050 + T^{2} \)
$23$ \( ( 28 + T )^{2} \)
$29$ \( ( -282 + T )^{2} \)
$31$ \( -75272 + T^{2} \)
$37$ \( ( -146 + T )^{2} \)
$41$ \( -116162 + T^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( -256328 + T^{2} \)
$53$ \( ( -598 + T )^{2} \)
$59$ \( -331298 + T^{2} \)
$61$ \( -217800 + T^{2} \)
$67$ \( ( 916 + T )^{2} \)
$71$ \( ( 420 + T )^{2} \)
$73$ \( -494018 + T^{2} \)
$79$ \( ( -292 + T )^{2} \)
$83$ \( -1328450 + T^{2} \)
$89$ \( -198450 + T^{2} \)
$97$ \( -347778 + T^{2} \)
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