Properties

Label 847.2.b.b
Level $847$
Weight $2$
Character orbit 847.b
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
CM discriminant -7
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: 8.0.37515625.1
Defining polynomial: \(x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 6 x^{3} - 4 x^{2} - 8 x + 16\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} + ( 1 - \beta_{2} + \beta_{7} ) q^{14} + ( 1 - 2 \beta_{2} - \beta_{6} + \beta_{7} ) q^{16} + 3 \beta_{1} q^{18} + ( 2 - \beta_{2} + \beta_{6} ) q^{23} + 5 q^{25} + ( 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{28} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{29} + ( 3 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{32} + ( -3 + 3 \beta_{2} ) q^{36} + ( -\beta_{2} - \beta_{6} + 2 \beta_{7} ) q^{37} + ( \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{43} + ( 5 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{46} -7 q^{49} + 5 \beta_{1} q^{50} + ( -3 + \beta_{2} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( -4 + 4 \beta_{2} + \beta_{6} ) q^{56} + ( -7 + 2 \beta_{2} + \beta_{7} ) q^{58} + ( -3 \beta_{1} + 3 \beta_{4} ) q^{63} + ( -5 + 5 \beta_{2} - \beta_{7} ) q^{64} + ( 1 - 3 \beta_{2} - \beta_{6} - 2 \beta_{7} ) q^{67} + ( 5 + \beta_{2} - \beta_{6} - 2 \beta_{7} ) q^{71} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{72} + ( 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{74} + ( -3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{79} + 9 q^{81} + ( 1 - 6 \beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{86} + ( -9 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{92} -7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 10q^{4} + 24q^{9} + O(q^{10}) \) \( 8q - 10q^{4} + 24q^{9} + 14q^{14} + 18q^{16} + 16q^{23} + 40q^{25} - 30q^{36} + 12q^{37} - 56q^{49} - 20q^{53} - 42q^{56} - 56q^{58} - 54q^{64} + 8q^{67} + 32q^{71} + 72q^{81} + 28q^{86} - 80q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 6 x^{3} - 4 x^{2} - 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - \nu^{5} + 3 \nu^{4} - \nu^{3} + 6 \nu^{2} + 4 \nu - 8 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + 3 \nu^{5} - \nu^{4} - 5 \nu^{3} + 4 \nu^{2} - 8 \nu + 8 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{6} + \nu^{5} - 3 \nu^{4} + 9 \nu^{3} + 16 \nu^{2} - 4 \nu + 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 17 \nu^{5} + 3 \nu^{4} - \nu^{3} + 6 \nu^{2} + 4 \nu - 96 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} - 7 \nu^{6} - \nu^{5} - \nu^{4} - \nu^{3} + 20 \nu^{2} - 38 \nu - 12 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -6 \nu^{7} - \nu^{6} + 7 \nu^{5} - 5 \nu^{4} + 3 \nu^{3} - 31 \nu^{2} - 4 \nu + 40 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -6 \nu^{7} - 3 \nu^{6} + 7 \nu^{5} - \nu^{4} + 3 \nu^{3} - 31 \nu^{2} - 26 \nu + 44 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{7} + 4 \beta_{6} - \beta_{2} + 11 \beta_{1} + 5\)\()/22\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 11 \beta_{2} + 8 \beta_{1} + 11\)\()/22\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + \beta_{6} + 11 \beta_{3} - 14 \beta_{2} + 11 \beta_{1} - 18\)\()/22\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{7} - 11 \beta_{6} - 5 \beta_{5} - \beta_{4} + 2 \beta_{3} + 29 \beta_{1} - 11\)\()/22\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{4} + \beta_{1} - 11\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(11 \beta_{7} - 22 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} - 63 \beta_{1} - 33\)\()/22\)
\(\nu^{7}\)\(=\)\((\)\(-8 \beta_{7} - 4 \beta_{6} + 11 \beta_{5} - 22 \beta_{4} - 33 \beta_{3} - 65 \beta_{2} - 44 \beta_{1} - 49\)\()/22\)

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
−0.373058 1.36412i
1.10362 0.884319i
1.18208 0.776336i
−1.41264 0.0667372i
−1.41264 + 0.0667372i
1.18208 + 0.776336i
1.10362 + 0.884319i
−0.373058 + 1.36412i
2.72824i 0 −5.44331 0 0 2.64575i 9.39419i 3.00000 0
846.2 1.76864i 0 −1.12808 0 0 2.64575i 1.54211i 3.00000 0
846.3 1.55267i 0 −0.410792 0 0 2.64575i 2.46752i 3.00000 0
846.4 0.133474i 0 1.98218 0 0 2.64575i 0.531520i 3.00000 0
846.5 0.133474i 0 1.98218 0 0 2.64575i 0.531520i 3.00000 0
846.6 1.55267i 0 −0.410792 0 0 2.64575i 2.46752i 3.00000 0
846.7 1.76864i 0 −1.12808 0 0 2.64575i 1.54211i 3.00000 0
846.8 2.72824i 0 −5.44331 0 0 2.64575i 9.39419i 3.00000 0
\(n\): 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 CM by \(\Q(\sqrt{-7}) \)
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.b 8
7.b odd 2 1 CM 847.2.b.b 8
11.b odd 2 1 inner 847.2.b.b 8
11.c even 5 1 77.2.l.a 8
11.c even 5 1 847.2.l.a 8
11.c even 5 1 847.2.l.c 8
11.c even 5 1 847.2.l.d 8
11.d odd 10 1 77.2.l.a 8
11.d odd 10 1 847.2.l.a 8
11.d odd 10 1 847.2.l.c 8
11.d odd 10 1 847.2.l.d 8
33.f even 10 1 693.2.bu.a 8
33.h odd 10 1 693.2.bu.a 8
77.b even 2 1 inner 847.2.b.b 8
77.j odd 10 1 77.2.l.a 8
77.j odd 10 1 847.2.l.a 8
77.j odd 10 1 847.2.l.c 8
77.j odd 10 1 847.2.l.d 8
77.l even 10 1 77.2.l.a 8
77.l even 10 1 847.2.l.a 8
77.l even 10 1 847.2.l.c 8
77.l even 10 1 847.2.l.d 8
77.m even 15 2 539.2.s.a 16
77.n even 30 2 539.2.s.a 16
77.o odd 30 2 539.2.s.a 16
77.p odd 30 2 539.2.s.a 16
231.r odd 10 1 693.2.bu.a 8
231.u even 10 1 693.2.bu.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.a 8 11.c even 5 1
77.2.l.a 8 11.d odd 10 1
77.2.l.a 8 77.j odd 10 1
77.2.l.a 8 77.l even 10 1
539.2.s.a 16 77.m even 15 2
539.2.s.a 16 77.n even 30 2
539.2.s.a 16 77.o odd 30 2
539.2.s.a 16 77.p odd 30 2
693.2.bu.a 8 33.f even 10 1
693.2.bu.a 8 33.h odd 10 1
693.2.bu.a 8 231.r odd 10 1
693.2.bu.a 8 231.u even 10 1
847.2.b.b 8 1.a even 1 1 trivial
847.2.b.b 8 7.b odd 2 1 CM
847.2.b.b 8 11.b odd 2 1 inner
847.2.b.b 8 77.b even 2 1 inner
847.2.l.a 8 11.c even 5 1
847.2.l.a 8 11.d odd 10 1
847.2.l.a 8 77.j odd 10 1
847.2.l.a 8 77.l even 10 1
847.2.l.c 8 11.c even 5 1
847.2.l.c 8 11.d odd 10 1
847.2.l.c 8 77.j odd 10 1
847.2.l.c 8 77.l even 10 1
847.2.l.d 8 11.c even 5 1
847.2.l.d 8 11.d odd 10 1
847.2.l.d 8 77.j odd 10 1
847.2.l.d 8 77.l even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 13 T_{2}^{6} + 49 T_{2}^{4} + 57 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 57 T^{2} + 49 T^{4} + 13 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 7 + T^{2} )^{4} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( -619 + 408 T - 51 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$29$ \( 259081 + 155562 T^{2} + 9499 T^{4} + 178 T^{6} + T^{8} \)
$31$ \( T^{8} \)
$37$ \( ( 1481 + 894 T - 149 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( 16072081 + 2478498 T^{2} + 53459 T^{4} + 402 T^{6} + T^{8} \)
$47$ \( T^{8} \)
$53$ \( ( -2455 - 1650 T - 165 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 17341 + 1276 T - 319 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$71$ \( ( -139 + 1584 T - 99 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( 23338561 + 3771482 T^{2} + 82859 T^{4} + 538 T^{6} + T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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