Properties

Label 784.4.a.w
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
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: \(1\)
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 196)
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.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} -25 q^{9} +O(q^{10})\) \( q + \beta q^{3} + 2 \beta q^{5} -25 q^{9} + 26 q^{11} + 24 \beta q^{13} + 4 q^{15} -73 \beta q^{17} -67 \beta q^{19} + 148 q^{23} -117 q^{25} -52 \beta q^{27} -118 q^{29} + 210 \beta q^{31} + 26 \beta q^{33} -254 q^{37} + 48 q^{39} + 65 \beta q^{41} -122 q^{43} -50 \beta q^{45} -218 \beta q^{47} -146 q^{51} -170 q^{53} + 52 \beta q^{55} -134 q^{57} -215 \beta q^{59} -430 \beta q^{61} + 96 q^{65} -420 q^{67} + 148 \beta q^{69} -420 q^{71} + 575 \beta q^{73} -117 \beta q^{75} -1052 q^{79} + 571 q^{81} + 1025 \beta q^{83} -292 q^{85} -118 \beta q^{87} -725 \beta q^{89} + 420 q^{93} -268 q^{95} + 223 \beta q^{97} -650 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 50q^{9} + O(q^{10}) \) \( 2q - 50q^{9} + 52q^{11} + 8q^{15} + 296q^{23} - 234q^{25} - 236q^{29} - 508q^{37} + 96q^{39} - 244q^{43} - 292q^{51} - 340q^{53} - 268q^{57} + 192q^{65} - 840q^{67} - 840q^{71} - 2104q^{79} + 1142q^{81} - 584q^{85} + 840q^{93} - 536q^{95} - 1300q^{99} + O(q^{100}) \)

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 −2.82843 0 0 0 −25.0000 0
1.2 0 1.41421 0 2.82843 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.w 2
4.b odd 2 1 196.4.a.f 2
7.b odd 2 1 inner 784.4.a.w 2
12.b even 2 1 1764.4.a.v 2
28.d even 2 1 196.4.a.f 2
28.f even 6 2 196.4.e.h 4
28.g odd 6 2 196.4.e.h 4
84.h odd 2 1 1764.4.a.v 2
84.j odd 6 2 1764.4.k.s 4
84.n even 6 2 1764.4.k.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.4.a.f 2 4.b odd 2 1
196.4.a.f 2 28.d even 2 1
196.4.e.h 4 28.f even 6 2
196.4.e.h 4 28.g odd 6 2
784.4.a.w 2 1.a even 1 1 trivial
784.4.a.w 2 7.b odd 2 1 inner
1764.4.a.v 2 12.b even 2 1
1764.4.a.v 2 84.h odd 2 1
1764.4.k.s 4 84.j odd 6 2
1764.4.k.s 4 84.n even 6 2

Hecke kernels

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

\( T_{3}^{2} - 2 \)
\( T_{5}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -26 + T )^{2} \)
$13$ \( -1152 + T^{2} \)
$17$ \( -10658 + T^{2} \)
$19$ \( -8978 + T^{2} \)
$23$ \( ( -148 + T )^{2} \)
$29$ \( ( 118 + T )^{2} \)
$31$ \( -88200 + T^{2} \)
$37$ \( ( 254 + T )^{2} \)
$41$ \( -8450 + T^{2} \)
$43$ \( ( 122 + T )^{2} \)
$47$ \( -95048 + T^{2} \)
$53$ \( ( 170 + T )^{2} \)
$59$ \( -92450 + T^{2} \)
$61$ \( -369800 + T^{2} \)
$67$ \( ( 420 + T )^{2} \)
$71$ \( ( 420 + T )^{2} \)
$73$ \( -661250 + T^{2} \)
$79$ \( ( 1052 + T )^{2} \)
$83$ \( -2101250 + T^{2} \)
$89$ \( -1051250 + T^{2} \)
$97$ \( -99458 + T^{2} \)
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