Properties

Label 784.4.a.bh
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 59 x^{2} + 60 x + 674\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 9 \beta_{1} - \beta_{3} ) q^{5} + ( 34 - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 9 \beta_{1} - \beta_{3} ) q^{5} + ( 34 - 3 \beta_{2} ) q^{9} + ( 29 + \beta_{2} ) q^{11} + ( -21 \beta_{1} - 9 \beta_{3} ) q^{13} + ( -28 - 8 \beta_{2} ) q^{15} + 65 \beta_{1} q^{17} + ( 6 \beta_{1} + 17 \beta_{3} ) q^{19} + ( 20 + 8 \beta_{2} ) q^{23} + ( 112 + 19 \beta_{2} ) q^{25} + ( 188 \beta_{1} + 16 \beta_{3} ) q^{27} + ( 9 - 7 \beta_{2} ) q^{29} + ( 88 \beta_{1} + 6 \beta_{3} ) q^{31} + 26 \beta_{3} q^{33} + ( 109 - 19 \beta_{2} ) q^{37} + ( -558 + 30 \beta_{2} ) q^{39} + ( -45 \beta_{1} + 36 \beta_{3} ) q^{41} + ( -93 - \beta_{2} ) q^{43} + ( 165 \beta_{1} + 23 \beta_{3} ) q^{45} + ( -212 \beta_{1} - 10 \beta_{3} ) q^{47} + ( 195 - 65 \beta_{2} ) q^{51} + ( 244 + 18 \beta_{2} ) q^{53} + ( 308 \beta_{1} - 48 \beta_{3} ) q^{55} + ( 953 - 23 \beta_{2} ) q^{57} + ( -498 \beta_{1} + 21 \beta_{3} ) q^{59} + ( 75 \beta_{1} + 25 \beta_{3} ) q^{61} + ( 195 + 69 \beta_{2} ) q^{65} + ( 294 + 22 \beta_{2} ) q^{67} + ( -424 \beta_{1} - 4 \beta_{3} ) q^{69} + ( 602 + 42 \beta_{2} ) q^{71} + ( 315 \beta_{1} + 42 \beta_{3} ) q^{73} + ( -878 \beta_{1} + 55 \beta_{3} ) q^{75} + ( -14 + 26 \beta_{2} ) q^{79} + ( 526 - 123 \beta_{2} ) q^{81} + ( 26 \beta_{1} + \beta_{3} ) q^{83} + ( 1235 + 65 \beta_{2} ) q^{85} + ( 424 \beta_{1} + 30 \beta_{3} ) q^{87} + ( -355 \beta_{1} + 16 \beta_{3} ) q^{89} + ( 594 - 94 \beta_{2} ) q^{93} + ( -1008 - 164 \beta_{2} ) q^{95} + ( 263 \beta_{1} + 22 \beta_{3} ) q^{97} + ( 647 - 53 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 136q^{9} + O(q^{10}) \) \( 4q + 136q^{9} + 116q^{11} - 112q^{15} + 80q^{23} + 448q^{25} + 36q^{29} + 436q^{37} - 2232q^{39} - 372q^{43} + 780q^{51} + 976q^{53} + 3812q^{57} + 780q^{65} + 1176q^{67} + 2408q^{71} - 56q^{79} + 2104q^{81} + 4940q^{85} + 2376q^{93} - 4032q^{95} + 2588q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 59 x^{2} + 60 x + 674\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} + 67 \nu - 34 \)\()/105\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 6 \nu^{2} + 344 \nu - 173 \)\()/105\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 51 \nu^{2} - 86 \nu - 1558 \)\()/105\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{3} + \beta_{2} + 61\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{3} + 35 \beta_{2} - 172 \beta_{1} + 91\)\()/2\)

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
−3.40086
4.40086
7.22929
−6.22929
0 −9.63797 0 −5.91838 0 0 0 65.8904 0
1.2 0 −5.39533 0 20.9517 0 0 0 2.10956 0
1.3 0 5.39533 0 −20.9517 0 0 0 2.10956 0
1.4 0 9.63797 0 5.91838 0 0 0 65.8904 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.bh 4
4.b odd 2 1 392.4.a.n 4
7.b odd 2 1 inner 784.4.a.bh 4
28.d even 2 1 392.4.a.n 4
28.f even 6 2 392.4.i.o 8
28.g odd 6 2 392.4.i.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.n 4 4.b odd 2 1
392.4.a.n 4 28.d even 2 1
392.4.i.o 8 28.f even 6 2
392.4.i.o 8 28.g odd 6 2
784.4.a.bh 4 1.a even 1 1 trivial
784.4.a.bh 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{4} - 122 T_{3}^{2} + 2704 \)
\( T_{5}^{4} - 474 T_{5}^{2} + 15376 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2704 - 122 T^{2} + T^{4} \)
$5$ \( 15376 - 474 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 728 - 58 T + T^{2} )^{2} \)
$13$ \( 16257024 - 10242 T^{2} + T^{4} \)
$17$ \( ( -8450 + T^{2} )^{2} \)
$19$ \( 266211856 - 32682 T^{2} + T^{4} \)
$23$ \( ( -6832 - 40 T + T^{2} )^{2} \)
$29$ \( ( -5456 - 18 T + T^{2} )^{2} \)
$31$ \( 154157056 - 32968 T^{2} + T^{4} \)
$37$ \( ( -28912 - 218 T + T^{2} )^{2} \)
$41$ \( 4262261796 - 162324 T^{2} + T^{4} \)
$43$ \( ( 8536 + 186 T + T^{2} )^{2} \)
$47$ \( 6407682304 - 182696 T^{2} + T^{4} \)
$53$ \( ( 22924 - 488 T + T^{2} )^{2} \)
$59$ \( 242288403984 - 1084122 T^{2} + T^{4} \)
$61$ \( 756250000 - 86250 T^{2} + T^{4} \)
$67$ \( ( 31744 - 588 T + T^{2} )^{2} \)
$71$ \( ( 163072 - 1204 T + T^{2} )^{2} \)
$73$ \( 5359118436 - 545076 T^{2} + T^{4} \)
$79$ \( ( -76192 + 28 T + T^{2} )^{2} \)
$83$ \( 1547536 - 2714 T^{2} + T^{4} \)
$89$ \( 62037857476 - 556004 T^{2} + T^{4} \)
$97$ \( 9932514244 - 308708 T^{2} + T^{4} \)
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