Properties

Label 784.4.f.h
Level $784$
Weight $4$
Character orbit 784.f
Analytic conductor $46.257$
Analytic rank $0$
Dimension $6$
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.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.12258833328.1
Defining polynomial: \(x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + ( 26 - \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 2 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + ( 26 - \beta_{1} - \beta_{2} ) q^{9} + ( -\beta_{3} - 11 \beta_{4} + \beta_{5} ) q^{11} + ( -\beta_{3} - 23 \beta_{4} ) q^{13} + ( 7 \beta_{3} - 19 \beta_{4} + \beta_{5} ) q^{15} + ( -5 \beta_{3} + 8 \beta_{5} ) q^{17} + ( -48 - \beta_{1} - 2 \beta_{2} ) q^{19} + ( 6 \beta_{3} - \beta_{5} ) q^{23} + ( -94 + 24 \beta_{2} ) q^{25} + ( 60 - 7 \beta_{1} - 8 \beta_{2} ) q^{27} + ( -50 - \beta_{1} + 23 \beta_{2} ) q^{29} + ( -76 - \beta_{1} - 24 \beta_{2} ) q^{31} + ( -4 \beta_{3} - 51 \beta_{4} + 8 \beta_{5} ) q^{33} + ( 75 - 2 \beta_{1} - 26 \beta_{2} ) q^{37} + ( -7 \beta_{3} - 27 \beta_{4} + 22 \beta_{5} ) q^{39} + ( -9 \beta_{3} + 141 \beta_{4} + 8 \beta_{5} ) q^{41} + ( -16 \beta_{3} + 20 \beta_{4} + 22 \beta_{5} ) q^{43} + ( 25 \beta_{3} - 219 \beta_{4} + 24 \beta_{5} ) q^{45} + ( -32 - \beta_{1} - 22 \beta_{2} ) q^{47} + ( -11 \beta_{3} - 289 \beta_{4} - 21 \beta_{5} ) q^{51} + ( 231 + 6 \beta_{1} + 54 \beta_{2} ) q^{53} + ( 146 + 16 \beta_{1} - 5 \beta_{2} ) q^{55} + ( 15 - 8 \beta_{1} + 40 \beta_{2} ) q^{57} + ( -106 - 2 \beta_{1} + \beta_{2} ) q^{59} + ( -6 \beta_{3} - 189 \beta_{4} + 8 \beta_{5} ) q^{61} + ( 196 + 23 \beta_{1} - \beta_{2} ) q^{65} + ( 42 \beta_{3} - 44 \beta_{4} - 25 \beta_{5} ) q^{67} + ( 39 \beta_{3} - 66 \beta_{4} + 8 \beta_{5} ) q^{69} + ( -14 \beta_{3} - 322 \beta_{4} - 26 \beta_{5} ) q^{71} + ( -42 \beta_{3} - 95 \beta_{4} - 8 \beta_{5} ) q^{73} + ( -1364 + 24 \beta_{1} + 70 \beta_{2} ) q^{75} + ( -14 \beta_{3} - 372 \beta_{4} + 47 \beta_{5} ) q^{79} + ( -99 - 23 \beta_{1} - 95 \beta_{2} ) q^{81} + ( 20 + 22 \beta_{1} + 70 \beta_{2} ) q^{83} + ( 599 + 40 \beta_{1} - 56 \beta_{2} ) q^{85} + ( -1214 + 17 \beta_{1} + 17 \beta_{2} ) q^{87} + ( 28 \beta_{3} - 291 \beta_{4} - 32 \beta_{5} ) q^{89} + ( 1037 - 30 \beta_{1} + 90 \beta_{2} ) q^{93} + ( -44 \beta_{3} - 238 \beta_{4} + 25 \beta_{5} ) q^{95} + ( -\beta_{3} + 45 \beta_{4} - 48 \beta_{5} ) q^{97} + ( 23 \beta_{3} - 113 \beta_{4} + 4 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 14q^{3} + 156q^{9} + O(q^{10}) \) \( 6q + 14q^{3} + 156q^{9} - 286q^{19} - 612q^{25} + 362q^{27} - 348q^{29} - 410q^{31} + 498q^{37} - 150q^{47} + 1290q^{53} + 918q^{55} - 6q^{57} - 642q^{59} + 1224q^{65} - 8276q^{75} - 450q^{81} + 24q^{83} + 3786q^{85} - 7284q^{87} + 5982q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 17 \nu^{5} - 493 \nu^{4} - 8463 \nu^{3} - 13736 \nu^{2} + 11424 \nu - 285776 \)\()/45520\)
\(\beta_{2}\)\(=\)\((\)\( -29 \nu^{5} + 841 \nu^{4} - 1629 \nu^{3} + 23432 \nu^{2} - 19488 \nu + 428592 \)\()/45520\)
\(\beta_{3}\)\(=\)\((\)\( -197 \nu^{5} + 23 \nu^{4} - 667 \nu^{3} - 5834 \nu^{2} + 117976 \nu - 52824 \)\()/68280\)
\(\beta_{4}\)\(=\)\((\)\( 203 \nu^{5} - 197 \nu^{4} + 5713 \nu^{3} + 986 \nu^{2} + 159176 \nu - 64104 \)\()/68280\)
\(\beta_{5}\)\(=\)\((\)\( 1191 \nu^{5} - 399 \nu^{4} + 34331 \nu^{3} - 17788 \nu^{2} + 982432 \nu - 404688 \)\()/45520\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_{1} + 2\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-12 \beta_{5} + 111 \beta_{4} + 3 \beta_{3} + 11 \beta_{2} - \beta_{1} - 110\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-17 \beta_{2} - 29 \beta_{1} - 22\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(116 \beta_{5} - 1029 \beta_{4} - 33 \beta_{3} + 105 \beta_{2} - 11 \beta_{1} - 1018\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(396 \beta_{5} - 1851 \beta_{4} - 2463 \beta_{3} + 425 \beta_{2} + 821 \beta_{1} + 1030\)\()/12\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
783.1
0.426664 0.739004i
0.426664 + 0.739004i
2.68858 4.65676i
2.68858 + 4.65676i
−2.61524 4.52973i
−2.61524 + 4.52973i
0 −7.06258 0 1.22397i 0 0 0 22.8800 0
783.2 0 −7.06258 0 1.22397i 0 0 0 22.8800 0
783.3 0 4.76834 0 16.8950i 0 0 0 −4.26295 0
783.4 0 4.76834 0 16.8950i 0 0 0 −4.26295 0
783.5 0 9.29424 0 19.8510i 0 0 0 59.3829 0
783.6 0 9.29424 0 19.8510i 0 0 0 59.3829 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 783.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.f.h 6
4.b odd 2 1 784.4.f.g 6
7.b odd 2 1 784.4.f.g 6
7.c even 3 1 112.4.p.f 6
7.d odd 6 1 112.4.p.g yes 6
28.d even 2 1 inner 784.4.f.h 6
28.f even 6 1 112.4.p.f 6
28.g odd 6 1 112.4.p.g yes 6
56.j odd 6 1 448.4.p.f 6
56.k odd 6 1 448.4.p.f 6
56.m even 6 1 448.4.p.g 6
56.p even 6 1 448.4.p.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.f 6 7.c even 3 1
112.4.p.f 6 28.f even 6 1
112.4.p.g yes 6 7.d odd 6 1
112.4.p.g yes 6 28.g odd 6 1
448.4.p.f 6 56.j odd 6 1
448.4.p.f 6 56.k odd 6 1
448.4.p.g 6 56.m even 6 1
448.4.p.g 6 56.p even 6 1
784.4.f.g 6 4.b odd 2 1
784.4.f.g 6 7.b odd 2 1
784.4.f.h 6 1.a even 1 1 trivial
784.4.f.h 6 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 7 T_{3}^{2} - 55 T_{3} + 313 \) acting on \(S_{4}^{\mathrm{new}}(784, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 313 - 55 T - 7 T^{2} + T^{3} )^{2} \)
$5$ \( 168507 + 113499 T^{2} + 681 T^{4} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 4876875 + 270279 T^{2} + 1821 T^{4} + T^{6} \)
$13$ \( 2167603200 + 7385616 T^{2} + 5304 T^{4} + T^{6} \)
$17$ \( 710833347 + 215397963 T^{2} + 29361 T^{4} + T^{6} \)
$19$ \( ( 33271 + 5537 T + 143 T^{2} + T^{3} )^{2} \)
$23$ \( 8915001507 + 129347991 T^{2} + 22677 T^{4} + T^{6} \)
$29$ \( ( -4622400 - 27504 T + 174 T^{2} + T^{3} )^{2} \)
$31$ \( ( -1532779 - 29071 T + 205 T^{2} + T^{3} )^{2} \)
$37$ \( ( 3813125 - 33525 T - 249 T^{2} + T^{3} )^{2} \)
$41$ \( 38591270836992 + 14002779024 T^{2} + 235176 T^{4} + T^{6} \)
$43$ \( 83474849412288 + 16477161840 T^{2} + 251796 T^{4} + T^{6} \)
$47$ \( ( -143829 - 34551 T + 75 T^{2} + T^{3} )^{2} \)
$53$ \( ( 61174089 - 116541 T - 645 T^{2} + T^{3} )^{2} \)
$59$ \( ( 908793 + 30753 T + 321 T^{2} + T^{3} )^{2} \)
$61$ \( 912934776095523 + 34600686795 T^{2} + 357585 T^{4} + T^{6} \)
$67$ \( 34861164967167075 + 360473787351 T^{2} + 1094421 T^{4} + T^{6} \)
$71$ \( 1026472074810048 + 436555595376 T^{2} + 1416948 T^{4} + T^{6} \)
$73$ \( 48340453761263403 + 565120567611 T^{2} + 1408473 T^{4} + T^{6} \)
$79$ \( 6567218609777523 + 1140744027447 T^{2} + 2151525 T^{4} + T^{6} \)
$83$ \( ( 308212992 - 856512 T - 12 T^{2} + T^{3} )^{2} \)
$89$ \( 6090737949537003 + 426701516283 T^{2} + 1389657 T^{4} + T^{6} \)
$97$ \( 19259496900926208 + 308008959120 T^{2} + 1038696 T^{4} + T^{6} \)
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