Properties

Label 784.4.a.bd
Level $784$
Weight $4$
Character orbit 784.a
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.1929.1
Defining polynomial: \(x^{3} - x^{2} - 10 x + 13\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -4 - \beta_{1} ) q^{5} + ( 15 + 3 \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -4 - \beta_{1} ) q^{5} + ( 15 + 3 \beta_{1} - 2 \beta_{2} ) q^{9} + ( 5 - \beta_{1} + 3 \beta_{2} ) q^{11} + ( -25 - \beta_{1} - 6 \beta_{2} ) q^{13} + ( 17 + \beta_{1} + 11 \beta_{2} ) q^{15} + ( -30 + 3 \beta_{1} + 10 \beta_{2} ) q^{17} + ( -21 - 7 \beta_{1} + 11 \beta_{2} ) q^{19} + ( -60 + 6 \beta_{1} + 17 \beta_{2} ) q^{23} + ( 6 + 2 \beta_{1} + 8 \beta_{2} ) q^{25} + ( 33 + 3 \beta_{1} - 13 \beta_{2} ) q^{27} + ( -61 + 7 \beta_{1} - 14 \beta_{2} ) q^{29} + ( 197 + 9 \beta_{1} + 3 \beta_{2} ) q^{31} + ( -109 - 8 \beta_{1} + 8 \beta_{2} ) q^{33} + ( 89 - 16 \beta_{1} + 34 \beta_{2} ) q^{37} + ( 269 + 19 \beta_{1} + 20 \beta_{2} ) q^{39} + ( -247 + 17 \beta_{1} - 26 \beta_{2} ) q^{41} + ( 92 + 8 \beta_{1} - 24 \beta_{2} ) q^{43} + ( -371 - 7 \beta_{1} - 2 \beta_{2} ) q^{45} + ( 31 + 27 \beta_{1} - 19 \beta_{2} ) q^{47} + ( -471 - 33 \beta_{1} + 29 \beta_{2} ) q^{51} + ( 71 + 10 \beta_{1} + 26 \beta_{2} ) q^{53} + ( 44 - 10 \beta_{1} - 25 \beta_{2} ) q^{55} + ( -343 - 26 \beta_{1} + 92 \beta_{2} ) q^{57} + ( 148 - 40 \beta_{1} + 51 \beta_{2} ) q^{59} + ( -165 + 48 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 317 + 29 \beta_{1} + 74 \beta_{2} ) q^{65} + ( 186 - 50 \beta_{1} - 11 \beta_{2} ) q^{67} + ( -816 - 57 \beta_{1} + 52 \beta_{2} ) q^{69} + ( -70 - 42 \beta_{1} - 28 \beta_{2} ) q^{71} + ( -581 - 14 \beta_{1} - 56 \beta_{2} ) q^{73} + ( -370 - 26 \beta_{1} - 4 \beta_{2} ) q^{75} + ( -428 + 58 \beta_{1} - 83 \beta_{2} ) q^{79} + ( 90 - 45 \beta_{1} - 26 \beta_{2} ) q^{81} + ( -458 + 22 \beta_{1} + 28 \beta_{2} ) q^{83} + ( -395 + 26 \beta_{1} - 134 \beta_{2} ) q^{85} + ( 469 + 35 \beta_{1} - 16 \beta_{2} ) q^{87} + ( -551 - 52 \beta_{1} + 44 \beta_{2} ) q^{89} + ( -279 - 18 \beta_{1} - 254 \beta_{2} ) q^{93} + ( 702 - 4 \beta_{1} - 65 \beta_{2} ) q^{95} + ( -877 + 31 \beta_{1} + 2 \beta_{2} ) q^{97} + ( -335 + 11 \beta_{1} + 100 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} - 13q^{5} + 50q^{9} + O(q^{10}) \) \( 3q + q^{3} - 13q^{5} + 50q^{9} + 11q^{11} - 70q^{13} + 41q^{15} - 97q^{17} - 81q^{19} - 191q^{23} + 12q^{25} + 115q^{27} - 162q^{29} + 597q^{31} - 343q^{33} + 217q^{37} + 806q^{39} - 698q^{41} + 308q^{43} - 1118q^{45} + 139q^{47} - 1475q^{51} + 197q^{53} + 147q^{55} - 1147q^{57} + 353q^{59} - 449q^{61} + 906q^{65} + 519q^{67} - 2557q^{69} - 224q^{71} - 1701q^{73} - 1132q^{75} - 1143q^{79} + 251q^{81} - 1380q^{83} - 1025q^{85} + 1458q^{87} - 1749q^{89} - 601q^{93} + 2167q^{95} - 2602q^{97} - 1094q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 10 x + 13\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} + 2 \nu - 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} - \beta_{1} + 29\)\()/4\)

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
2.90222
−3.27144
1.36922
0 −7.65024 0 −14.6089 0 0 0 31.5262 0
1.2 0 0.138208 0 10.0858 0 0 0 −26.9809 0
1.3 0 8.51203 0 −8.47688 0 0 0 45.4547 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.bd 3
4.b odd 2 1 392.4.a.j 3
7.b odd 2 1 784.4.a.bc 3
7.d odd 6 2 112.4.i.f 6
28.d even 2 1 392.4.a.k 3
28.f even 6 2 56.4.i.a 6
28.g odd 6 2 392.4.i.n 6
56.j odd 6 2 448.4.i.k 6
56.m even 6 2 448.4.i.l 6
84.j odd 6 2 504.4.s.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.i.a 6 28.f even 6 2
112.4.i.f 6 7.d odd 6 2
392.4.a.j 3 4.b odd 2 1
392.4.a.k 3 28.d even 2 1
392.4.i.n 6 28.g odd 6 2
448.4.i.k 6 56.j odd 6 2
448.4.i.l 6 56.m even 6 2
504.4.s.i 6 84.j odd 6 2
784.4.a.bc 3 7.b odd 2 1
784.4.a.bd 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{3} - T_{3}^{2} - 65 T_{3} + 9 \)
\( T_{5}^{3} + 13 T_{5}^{2} - 109 T_{5} - 1249 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 9 - 65 T - T^{2} + T^{3} \)
$5$ \( -1249 - 109 T + 13 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 8099 - 577 T - 11 T^{2} + T^{3} \)
$13$ \( -17752 - 1156 T + 70 T^{2} + T^{3} \)
$17$ \( -586557 - 6245 T + 97 T^{2} + T^{3} \)
$19$ \( -123377 - 10329 T + 81 T^{2} + T^{3} \)
$23$ \( -3492279 - 17297 T + 191 T^{2} + T^{3} \)
$29$ \( -1324296 - 7716 T + 162 T^{2} + T^{3} \)
$31$ \( -4663139 + 103599 T - 597 T^{2} + T^{3} \)
$37$ \( 15110373 - 77493 T - 217 T^{2} + T^{3} \)
$41$ \( -6465192 + 90492 T + 698 T^{2} + T^{3} \)
$43$ \( -38848 - 7888 T - 308 T^{2} + T^{3} \)
$47$ \( 18709731 - 114417 T - 139 T^{2} + T^{3} \)
$53$ \( -2914839 - 59549 T - 197 T^{2} + T^{3} \)
$59$ \( 37291113 - 300449 T - 353 T^{2} + T^{3} \)
$61$ \( 9942147 - 318341 T + 449 T^{2} + T^{3} \)
$67$ \( 21323807 - 356385 T - 519 T^{2} + T^{3} \)
$71$ \( -7551488 - 379456 T + 224 T^{2} + T^{3} \)
$73$ \( 72721831 + 691635 T + 1701 T^{2} + T^{3} \)
$79$ \( -297161663 - 352545 T + 1143 T^{2} + T^{3} \)
$83$ \( 4797376 + 475632 T + 1380 T^{2} + T^{3} \)
$89$ \( -155620521 + 549843 T + 1749 T^{2} + T^{3} \)
$97$ \( 528741144 + 2094844 T + 2602 T^{2} + T^{3} \)
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