Properties

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

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: \(0\)
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: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 3 \beta_{1} + \beta_{3} ) q^{5} + ( 10 - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 3 \beta_{1} + \beta_{3} ) q^{5} + ( 10 - 3 \beta_{2} ) q^{9} + ( 31 - 5 \beta_{2} ) q^{11} + ( \beta_{1} + 9 \beta_{3} ) q^{13} + ( 40 - 4 \beta_{2} ) q^{15} + ( 31 \beta_{1} + 8 \beta_{3} ) q^{17} + ( -2 \beta_{1} + \beta_{3} ) q^{19} + ( -32 + 20 \beta_{2} ) q^{23} + ( -80 - 5 \beta_{2} ) q^{25} + ( 68 \beta_{1} - 8 \beta_{3} ) q^{27} + ( -63 + 9 \beta_{2} ) q^{29} + ( -64 \beta_{1} + 14 \beta_{3} ) q^{31} + ( 232 \beta_{1} + 46 \beta_{3} ) q^{33} + ( -91 + 29 \beta_{2} ) q^{37} + ( 282 - 10 \beta_{2} ) q^{39} + ( -291 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 305 + 21 \beta_{2} ) q^{43} + ( 135 \beta_{1} + 25 \beta_{3} ) q^{45} + ( 172 \beta_{1} + 70 \beta_{3} ) q^{47} + ( 341 - 39 \beta_{2} ) q^{51} + ( -188 - 46 \beta_{2} ) q^{53} + ( 268 \beta_{1} + 56 \beta_{3} ) q^{55} + ( 25 + \beta_{2} ) q^{57} + ( -250 \beta_{1} + 29 \beta_{3} ) q^{59} + ( 201 \beta_{1} - 73 \beta_{3} ) q^{61} + ( 275 - 19 \beta_{2} ) q^{65} + ( 686 - 18 \beta_{2} ) q^{67} + ( -744 \beta_{1} - 92 \beta_{3} ) q^{69} + ( 430 + 46 \beta_{2} ) q^{71} + ( -83 \beta_{1} - 50 \beta_{3} ) q^{73} + ( 10 \beta_{1} - 65 \beta_{3} ) q^{75} + ( 422 - 2 \beta_{2} ) q^{79} + ( -314 + 21 \beta_{2} ) q^{81} + ( 106 \beta_{1} - 23 \beta_{3} ) q^{83} + ( 395 - 47 \beta_{2} ) q^{85} + ( -432 \beta_{1} - 90 \beta_{3} ) q^{87} + ( 611 \beta_{1} - 8 \beta_{3} ) q^{89} + ( 242 + 50 \beta_{2} ) q^{93} + 20 q^{95} + ( 177 \beta_{1} + 210 \beta_{3} ) q^{97} + ( 1285 - 143 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 40q^{9} + O(q^{10}) \) \( 4q + 40q^{9} + 124q^{11} + 160q^{15} - 128q^{23} - 320q^{25} - 252q^{29} - 364q^{37} + 1128q^{39} + 1220q^{43} + 1364q^{51} - 752q^{53} + 100q^{57} + 1100q^{65} + 2744q^{67} + 1720q^{71} + 1688q^{79} - 1256q^{81} + 1580q^{85} + 968q^{93} + 80q^{95} + 5140q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} + 3 \nu^{2} + 43 \nu - 22 \)\()/57\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{3} + 6 \nu^{2} + 200 \nu - 101 \)\()/57\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 27 \nu^{2} - 50 \nu - 502 \)\()/57\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(4 \beta_{3} + \beta_{2} + 37\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{3} + 23 \beta_{2} - 100 \beta_{1} + 55\)\()/2\)

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 −9.23641 0 0 0 34.1868 0
1.2 0 −3.57956 0 −2.16534 0 0 0 −14.1868 0
1.3 0 3.57956 0 2.16534 0 0 0 −14.1868 0
1.4 0 7.82220 0 9.23641 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.bg 4
4.b odd 2 1 392.4.a.m 4
7.b odd 2 1 inner 784.4.a.bg 4
28.d even 2 1 392.4.a.m 4
28.f even 6 2 392.4.i.p 8
28.g odd 6 2 392.4.i.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.m 4 4.b odd 2 1
392.4.a.m 4 28.d even 2 1
392.4.i.p 8 28.f even 6 2
392.4.i.p 8 28.g odd 6 2
784.4.a.bg 4 1.a even 1 1 trivial
784.4.a.bg 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(784))\):

\( T_{3}^{4} - 74 T_{3}^{2} + 784 \)
\( T_{5}^{4} - 90 T_{5}^{2} + 400 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 784 - 74 T^{2} + T^{4} \)
$5$ \( 400 - 90 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -664 - 62 T + T^{2} )^{2} \)
$13$ \( 6801664 - 5314 T^{2} + T^{4} \)
$17$ \( 386884 - 7076 T^{2} + T^{4} \)
$19$ \( 400 - 90 T^{2} + T^{4} \)
$23$ \( ( -24976 + 64 T + T^{2} )^{2} \)
$29$ \( ( -1296 + 126 T + T^{2} )^{2} \)
$31$ \( 13778944 - 32904 T^{2} + T^{4} \)
$37$ \( ( -46384 + 182 T + T^{2} )^{2} \)
$41$ \( 27729576484 - 335124 T^{2} + T^{4} \)
$43$ \( ( 64360 - 610 T + T^{2} )^{2} \)
$47$ \( 14813810944 - 393576 T^{2} + T^{4} \)
$53$ \( ( -102196 + 376 T + T^{2} )^{2} \)
$59$ \( 12676057744 - 334506 T^{2} + T^{4} \)
$61$ \( 3645744400 - 572010 T^{2} + T^{4} \)
$67$ \( ( 449536 - 1372 T + T^{2} )^{2} \)
$71$ \( ( 47360 - 860 T + T^{2} )^{2} \)
$73$ \( 5553528484 - 175956 T^{2} + T^{4} \)
$79$ \( ( 177824 - 844 T + T^{2} )^{2} \)
$83$ \( 108576400 - 89610 T^{2} + T^{4} \)
$89$ \( 569074096900 - 1517060 T^{2} + T^{4} \)
$97$ \( 2024593185924 - 2887236 T^{2} + T^{4} \)
show more
show less