Properties

Label 847.2.b.d
Level $847$
Weight $2$
Character orbit 847.b
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.77720518656.2
Defining polynomial: \(x^{8} - 4 x^{7} + 26 x^{6} - 64 x^{5} + 161 x^{4} - 220 x^{3} + 232 x^{2} - 132 x + 33\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} -\beta_{1} q^{4} + \beta_{7} q^{5} + ( -\beta_{3} - \beta_{5} - \beta_{6} ) q^{7} + ( \beta_{5} + \beta_{6} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q + \beta_{6} q^{2} -\beta_{1} q^{4} + \beta_{7} q^{5} + ( -\beta_{3} - \beta_{5} - \beta_{6} ) q^{7} + ( \beta_{5} + \beta_{6} ) q^{8} + 3 q^{9} + ( \beta_{3} - \beta_{4} ) q^{10} + ( -\beta_{3} + \beta_{4} ) q^{13} + ( 1 + \beta_{2} + \beta_{7} ) q^{14} + ( -1 - 2 \beta_{1} ) q^{16} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{17} + 3 \beta_{6} q^{18} + ( -2 \beta_{3} - \beta_{5} ) q^{19} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{7} ) q^{20} + ( -3 - \beta_{1} ) q^{23} -6 q^{25} + ( -1 - \beta_{1} + 2 \beta_{2} + 3 \beta_{7} ) q^{26} + ( -\beta_{4} - 2 \beta_{5} ) q^{28} + ( 4 \beta_{5} + \beta_{6} ) q^{29} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( 4 \beta_{5} - \beta_{6} ) q^{32} + \beta_{7} q^{34} + ( \beta_{4} - 5 \beta_{5} ) q^{35} -3 \beta_{1} q^{36} + ( -5 + 2 \beta_{1} ) q^{37} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{7} ) q^{38} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{40} + ( \beta_{3} - \beta_{4} ) q^{41} + ( -2 \beta_{5} - 6 \beta_{6} ) q^{43} + 3 \beta_{7} q^{45} + ( \beta_{5} - 4 \beta_{6} ) q^{46} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{47} + ( 5 + \beta_{1} - 2 \beta_{2} - \beta_{7} ) q^{49} -6 \beta_{6} q^{50} + ( 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{52} + ( -3 - 4 \beta_{1} ) q^{53} + ( 1 - \beta_{1} + \beta_{2} ) q^{56} + ( 2 + 3 \beta_{1} ) q^{58} -2 \beta_{7} q^{59} + ( -2 \beta_{4} - \beta_{5} ) q^{61} + ( -2 \beta_{3} - \beta_{5} ) q^{62} + ( -3 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} ) q^{63} + ( 4 + \beta_{1} ) q^{64} + 11 \beta_{6} q^{65} + ( -1 - 5 \beta_{1} ) q^{67} + ( 3 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{68} + ( -6 - 6 \beta_{1} + \beta_{2} + 2 \beta_{7} ) q^{70} + ( -4 + 2 \beta_{1} ) q^{71} + ( 3 \beta_{5} + 3 \beta_{6} ) q^{72} + ( 2 \beta_{3} + \beta_{5} ) q^{73} + ( -2 \beta_{5} - 3 \beta_{6} ) q^{74} + ( -2 \beta_{4} - \beta_{5} ) q^{76} + ( 5 \beta_{5} + 4 \beta_{6} ) q^{79} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{7} ) q^{80} + 9 q^{81} + ( 1 + \beta_{1} - 2 \beta_{2} - 3 \beta_{7} ) q^{82} + ( 4 \beta_{3} + 2 \beta_{5} ) q^{83} + ( 11 \beta_{5} + 11 \beta_{6} ) q^{85} + ( 10 + 4 \beta_{1} ) q^{86} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{7} ) q^{89} + ( 3 \beta_{3} - 3 \beta_{4} ) q^{90} + ( 6 + 6 \beta_{1} - \beta_{2} - 2 \beta_{7} ) q^{91} + ( 3 + 3 \beta_{1} ) q^{92} + ( 2 \beta_{3} + \beta_{5} ) q^{94} -11 \beta_{5} q^{95} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{7} ) q^{97} + ( -3 \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} + 12q^{14} - 8q^{16} - 24q^{23} - 48q^{25} - 40q^{37} + 32q^{49} - 24q^{53} + 12q^{56} + 16q^{58} + 32q^{64} - 8q^{67} - 44q^{70} - 32q^{71} + 72q^{81} + 80q^{86} + 44q^{91} + 24q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 26 x^{6} - 64 x^{5} + 161 x^{4} - 220 x^{3} + 232 x^{2} - 132 x + 33\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} - 6 \nu^{5} + 47 \nu^{4} - 84 \nu^{3} + 239 \nu^{2} - 198 \nu + 138 \)\()/45\)
\(\beta_{2}\)\(=\)\((\)\( 8 \nu^{7} + 11 \nu^{6} + 38 \nu^{5} + 599 \nu^{4} - 1864 \nu^{3} + 5303 \nu^{2} - 12048 \nu + 7545 \)\()/1755\)
\(\beta_{3}\)\(=\)\((\)\( -35 \nu^{7} - 53 \nu^{6} - 371 \nu^{5} - 1748 \nu^{4} + 901 \nu^{3} - 8756 \nu^{2} + 4272 \nu - 4827 \)\()/1755\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{7} + 5 \nu^{6} - 121 \nu^{5} + 17 \nu^{4} - 475 \nu^{3} + 464 \nu^{2} - 1041 \nu + 1866 \)\()/351\)
\(\beta_{5}\)\(=\)\((\)\( 14 \nu^{7} - 49 \nu^{6} + 359 \nu^{5} - 775 \nu^{4} + 2237 \nu^{3} - 2605 \nu^{2} + 3135 \nu - 1158 \)\()/351\)
\(\beta_{6}\)\(=\)\((\)\( -188 \nu^{7} + 658 \nu^{6} - 4520 \nu^{5} + 9655 \nu^{4} - 24524 \nu^{3} + 27460 \nu^{2} - 26103 \nu + 8781 \)\()/1755\)
\(\beta_{7}\)\(=\)\((\)\( -376 \nu^{7} + 1316 \nu^{6} - 9040 \nu^{5} + 19310 \nu^{4} - 49048 \nu^{3} + 54920 \nu^{2} - 48696 \nu + 15807 \)\()/1755\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{1} - 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-11 \beta_{7} + 21 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - 11\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-23 \beta_{7} + 44 \beta_{6} - 20 \beta_{4} + 28 \beta_{3} - 12 \beta_{2} + 44 \beta_{1} + 87\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(116 \beta_{7} - 207 \beta_{6} + 68 \beta_{5} - 55 \beta_{4} + 75 \beta_{3} + 80 \beta_{2} + 60 \beta_{1} + 186\)\()/2\)
\(\nu^{6}\)\(=\)\(203 \beta_{7} - 366 \beta_{6} + 60 \beta_{5} + 96 \beta_{4} - 160 \beta_{3} + 135 \beta_{2} - 285 \beta_{1} - 456\)
\(\nu^{7}\)\(=\)\((\)\(-1107 \beta_{7} + 1902 \beta_{6} - 954 \beta_{5} + 868 \beta_{4} - 1386 \beta_{3} - 812 \beta_{2} - 1470 \beta_{1} - 3121\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
846.1
0.500000 + 0.273539i
0.500000 + 3.59016i
0.500000 1.14067i
0.500000 + 2.17595i
0.500000 2.17595i
0.500000 + 1.14067i
0.500000 3.59016i
0.500000 0.273539i
1.93185i 0 −1.73205 3.31662i 0 2.34521 + 1.22474i 0.517638i 3.00000 −6.40723
846.2 1.93185i 0 −1.73205 3.31662i 0 −2.34521 + 1.22474i 0.517638i 3.00000 6.40723
846.3 0.517638i 0 1.73205 3.31662i 0 −2.34521 + 1.22474i 1.93185i 3.00000 −1.71681
846.4 0.517638i 0 1.73205 3.31662i 0 2.34521 + 1.22474i 1.93185i 3.00000 1.71681
846.5 0.517638i 0 1.73205 3.31662i 0 2.34521 1.22474i 1.93185i 3.00000 1.71681
846.6 0.517638i 0 1.73205 3.31662i 0 −2.34521 1.22474i 1.93185i 3.00000 −1.71681
846.7 1.93185i 0 −1.73205 3.31662i 0 −2.34521 1.22474i 0.517638i 3.00000 6.40723
846.8 1.93185i 0 −1.73205 3.31662i 0 2.34521 1.22474i 0.517638i 3.00000 −6.40723
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 846.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.b.d 8
7.b odd 2 1 inner 847.2.b.d 8
11.b odd 2 1 inner 847.2.b.d 8
11.c even 5 4 847.2.l.k 32
11.d odd 10 4 847.2.l.k 32
77.b even 2 1 inner 847.2.b.d 8
77.j odd 10 4 847.2.l.k 32
77.l even 10 4 847.2.l.k 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.b.d 8 1.a even 1 1 trivial
847.2.b.d 8 7.b odd 2 1 inner
847.2.b.d 8 11.b odd 2 1 inner
847.2.b.d 8 77.b even 2 1 inner
847.2.l.k 32 11.c even 5 4
847.2.l.k 32 11.d odd 10 4
847.2.l.k 32 77.j odd 10 4
847.2.l.k 32 77.l even 10 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( ( 11 + T^{2} )^{4} \)
$7$ \( ( 49 - 8 T^{2} + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( ( 121 - 44 T^{2} + T^{4} )^{2} \)
$17$ \( ( 121 - 44 T^{2} + T^{4} )^{2} \)
$19$ \( ( -22 + T^{2} )^{4} \)
$23$ \( ( 6 + 6 T + T^{2} )^{4} \)
$29$ \( ( 529 + 52 T^{2} + T^{4} )^{2} \)
$31$ \( ( 484 + 88 T^{2} + T^{4} )^{2} \)
$37$ \( ( 13 + 10 T + T^{2} )^{4} \)
$41$ \( ( 121 - 44 T^{2} + T^{4} )^{2} \)
$43$ \( ( 2704 + 112 T^{2} + T^{4} )^{2} \)
$47$ \( ( 484 + 88 T^{2} + T^{4} )^{2} \)
$53$ \( ( -39 + 6 T + T^{2} )^{4} \)
$59$ \( ( 44 + T^{2} )^{4} \)
$61$ \( ( -66 + T^{2} )^{4} \)
$67$ \( ( -74 + 2 T + T^{2} )^{4} \)
$71$ \( ( 4 + 8 T + T^{2} )^{4} \)
$73$ \( ( -22 + T^{2} )^{4} \)
$79$ \( ( 36 + 84 T^{2} + T^{4} )^{2} \)
$83$ \( ( -88 + T^{2} )^{4} \)
$89$ \( ( 33 + T^{2} )^{4} \)
$97$ \( ( 14641 + 286 T^{2} + T^{4} )^{2} \)
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