Properties

Label 784.4.a.bf
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( 10 - 3 \beta_{1} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( 10 - 3 \beta_{1} ) q^{9} + ( -25 + 3 \beta_{1} ) q^{11} + ( 6 \beta_{2} - \beta_{3} ) q^{13} + ( 16 + 4 \beta_{1} ) q^{15} + ( -\beta_{2} - 9 \beta_{3} ) q^{17} + ( 11 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -88 - 4 \beta_{1} ) q^{23} + ( -32 + 11 \beta_{1} ) q^{25} + ( -4 \beta_{2} - 12 \beta_{3} ) q^{27} + ( 65 - 7 \beta_{1} ) q^{29} + ( 38 \beta_{2} + 4 \beta_{3} ) q^{31} + ( 46 \beta_{2} + 12 \beta_{3} ) q^{33} + ( 53 - 3 \beta_{1} ) q^{37} + ( -238 + 14 \beta_{1} ) q^{39} + ( -31 \beta_{2} - 3 \beta_{3} ) q^{41} + ( -135 - 35 \beta_{1} ) q^{43} + ( 12 \beta_{2} - 11 \beta_{3} ) q^{45} + ( 26 \beta_{2} + 32 \beta_{3} ) q^{47} + ( -107 - 39 \beta_{1} ) q^{51} + ( 4 + 18 \beta_{1} ) q^{53} + ( -12 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -471 + 17 \beta_{1} ) q^{57} + ( -49 \beta_{2} - 20 \beta_{3} ) q^{59} + ( 68 \beta_{2} + 27 \beta_{3} ) q^{61} + ( -189 - 35 \beta_{1} ) q^{65} + ( 486 + 38 \beta_{1} ) q^{67} + ( 60 \beta_{2} - 16 \beta_{3} ) q^{69} + ( -562 + 14 \beta_{1} ) q^{71} + ( -37 \beta_{2} + 25 \beta_{3} ) q^{73} + ( 109 \beta_{2} + 44 \beta_{3} ) q^{75} + ( 262 + 94 \beta_{1} ) q^{79} + ( -314 + 21 \beta_{1} ) q^{81} + ( -\beta_{2} - 8 \beta_{3} ) q^{83} + ( -821 - 95 \beta_{1} ) q^{85} + ( -114 \beta_{2} - 28 \beta_{3} ) q^{87} + ( -157 \beta_{2} + 75 \beta_{3} ) q^{89} + ( -1342 + 130 \beta_{1} ) q^{93} + ( -548 - 88 \beta_{1} ) q^{95} + ( -189 \beta_{2} - 91 \beta_{3} ) q^{97} + ( -835 + 105 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 40q^{9} + O(q^{10}) \) \( 4q + 40q^{9} - 100q^{11} + 64q^{15} - 352q^{23} - 128q^{25} + 260q^{29} + 212q^{37} - 952q^{39} - 540q^{43} - 428q^{51} + 16q^{53} - 1884q^{57} - 756q^{65} + 1944q^{67} - 2248q^{71} + 1048q^{79} - 1256q^{81} - 3284q^{85} - 5368q^{93} - 2192q^{95} - 3340q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4 \nu^{3} + 6 \nu^{2} + 200 \nu - 101 \)\()/57\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 11 \nu^{2} - 12 \nu + 182 \)\()/19\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{3} + 12 \nu^{2} - 265 \nu - 392 \)\()/57\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 7\)\()/14\)
\(\nu^{2}\)\(=\)\((\)\(8 \beta_{3} - 20 \beta_{2} + 7 \beta_{1} + 259\)\()/14\)
\(\nu^{3}\)\(=\)\((\)\(16 \beta_{3} + 10 \beta_{2} + 23 \beta_{1} + 55\)\()/2\)

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
−2.11692
3.11692
5.94534
−4.94534
0 −7.82220 0 2.07730 0 0 0 34.1868 0
1.2 0 −3.57956 0 −13.4791 0 0 0 −14.1868 0
1.3 0 3.57956 0 13.4791 0 0 0 −14.1868 0
1.4 0 7.82220 0 −2.07730 0 0 0 34.1868 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.bf 4
4.b odd 2 1 49.4.a.e 4
7.b odd 2 1 inner 784.4.a.bf 4
12.b even 2 1 441.4.a.u 4
20.d odd 2 1 1225.4.a.bb 4
28.d even 2 1 49.4.a.e 4
28.f even 6 2 49.4.c.e 8
28.g odd 6 2 49.4.c.e 8
84.h odd 2 1 441.4.a.u 4
84.j odd 6 2 441.4.e.y 8
84.n even 6 2 441.4.e.y 8
140.c even 2 1 1225.4.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 4.b odd 2 1
49.4.a.e 4 28.d even 2 1
49.4.c.e 8 28.f even 6 2
49.4.c.e 8 28.g odd 6 2
441.4.a.u 4 12.b even 2 1
441.4.a.u 4 84.h odd 2 1
441.4.e.y 8 84.j odd 6 2
441.4.e.y 8 84.n even 6 2
784.4.a.bf 4 1.a even 1 1 trivial
784.4.a.bf 4 7.b odd 2 1 inner
1225.4.a.bb 4 20.d odd 2 1
1225.4.a.bb 4 140.c even 2 1

Hecke kernels

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

\( T_{3}^{4} - 74 T_{3}^{2} + 784 \)
\( T_{5}^{4} - 186 T_{5}^{2} + 784 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 784 - 74 T^{2} + T^{4} \)
$5$ \( 784 - 186 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 40 + 50 T + T^{2} )^{2} \)
$13$ \( 2458624 - 3234 T^{2} + T^{4} \)
$17$ \( 9746884 - 14564 T^{2} + T^{4} \)
$19$ \( 52591504 - 14746 T^{2} + T^{4} \)
$23$ \( ( 6704 + 176 T + T^{2} )^{2} \)
$29$ \( ( 1040 - 130 T + T^{2} )^{2} \)
$31$ \( 629407744 - 100104 T^{2} + T^{4} \)
$37$ \( ( 2224 - 106 T + T^{2} )^{2} \)
$41$ \( 307721764 - 66836 T^{2} + T^{4} \)
$43$ \( ( -61400 + 270 T + T^{2} )^{2} \)
$47$ \( 8332038400 - 187240 T^{2} + T^{4} \)
$53$ \( ( -21044 - 8 T + T^{2} )^{2} \)
$59$ \( 1600960144 - 189354 T^{2} + T^{4} \)
$61$ \( 5022273424 - 360266 T^{2} + T^{4} \)
$67$ \( ( 142336 - 972 T + T^{2} )^{2} \)
$71$ \( ( 303104 + 1124 T + T^{2} )^{2} \)
$73$ \( 12428236324 - 276756 T^{2} + T^{4} \)
$79$ \( ( -505696 - 524 T + T^{2} )^{2} \)
$83$ \( 6492304 - 11466 T^{2} + T^{4} \)
$89$ \( 2844693770884 - 3623876 T^{2} + T^{4} \)
$97$ \( 841222821124 - 3082884 T^{2} + T^{4} \)
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