Newform orbit 847.2.f.r

• Label: 847.2.f.r
• Level: $847$
• Weight: $2$
• Character orbit: 847.f
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.f (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.446265625.1
Defining polynomial:	\( x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{3} + (\beta_{3} - \beta_1) q^{4} + (2 \beta_{6} + \beta_{4}) q^{5} + (\beta_{6} - 3 \beta_{4}) q^{6} + \beta_{3} q^{7} + (3 \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + 3) q^{8} + \beta_{5} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + q^{3} - 3 q^{4} - 7 q^{6} - 2 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 4x^{6} - 7x^{5} + 19x^{4} + 21x^{3} + 36x^{2} + 27x + 81 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} - 21 ) / 19 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{6} + 40\nu ) / 57 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} - 97\nu^{2} ) / 171 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 4\nu^{6} - 7\nu^{5} + 19\nu^{4} - 76\nu^{3} + 36\nu^{2} + 27\nu + 81 ) / 171 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{7} - 40\nu^{2} ) / 57 \)
\(\beta_{7}\)	\(=\)	\( ( 4\nu^{7} - 16\nu^{6} + 28\nu^{5} - 76\nu^{4} + 133\nu^{3} - 144\nu^{2} - 108\nu - 324 ) / 513 \)

Character values

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

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(1\)	\(\beta_{3}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

148.1

1.86298	−	1.35354i
−1.05397	+	0.765752i
0.402580	+	1.23901i
−0.711597	−	2.19007i
1.86298	+	1.35354i
−1.05397	−	0.765752i
0.402580	−	1.23901i
−0.711597	+	2.19007i

−0.711597

−

2.19007i

1.86298

+

1.35354i

−2.67200

+

1.94132i

1.11418

−

3.42908i

1.63865

−

5.04324i

−0.809017

+

0.587785i

2.42705

+

1.76336i

0.711597

+

2.19007i

−8.30278

148.2

0.402580

+

1.23901i

−1.05397

−

0.765752i

0.244951

−

0.177967i

−1.11418

+

3.42908i

0.524471

−

1.61416i

−0.809017

+

0.587785i

2.42705

+

1.76336i

−0.402580

−

1.23901i

−4.69722

323.1

−1.05397

−

0.765752i

0.402580

−

1.23901i

−0.0935628

−

0.287957i

2.91695

−

2.11929i

−1.37308

+

0.997603i

0.309017

+

0.951057i

−0.927051

+

2.85317i

1.05397

+

0.765752i

−4.69722

323.2

1.86298

+

1.35354i

−0.711597

+

2.19007i

1.02061

+

3.14113i

−2.91695

+

2.11929i

−4.29004

+

3.11689i

0.309017

+

0.951057i

−0.927051

+

2.85317i

−1.86298

−

1.35354i

−8.30278

372.1

−0.711597

+

2.19007i

1.86298

−

1.35354i

−2.67200

−

1.94132i

1.11418

+

3.42908i

1.63865

+

5.04324i

−0.809017

−

0.587785i

2.42705

−

1.76336i

0.711597

−

2.19007i

−8.30278

372.2

0.402580

−

1.23901i

−1.05397

+

0.765752i

0.244951

+

0.177967i

−1.11418

−

3.42908i

0.524471

+

1.61416i

−0.809017

−

0.587785i

2.42705

−

1.76336i

−0.402580

+

1.23901i

−4.69722

729.1

−1.05397

+

0.765752i

0.402580

+

1.23901i

−0.0935628

+

0.287957i

2.91695

+

2.11929i

−1.37308

−

0.997603i

0.309017

−

0.951057i

−0.927051

−

2.85317i

1.05397

−

0.765752i

−4.69722

729.2

1.86298

−

1.35354i

−0.711597

−

2.19007i

1.02061

−

3.14113i

−2.91695

−

2.11929i

−4.29004

−

3.11689i

0.309017

−

0.951057i

−0.927051

−

2.85317i

−1.86298

+

1.35354i

−8.30278

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.2.f.r		8
11.b	odd	2	1		847.2.f.o		8
11.c	even	5	1		847.2.a.e	✓	2
11.c	even	5	3	inner	847.2.f.r		8
11.d	odd	10	1		847.2.a.g	yes	2
11.d	odd	10	3		847.2.f.o		8
33.f	even	10	1		7623.2.a.bc		2
33.h	odd	10	1		7623.2.a.bs		2
77.j	odd	10	1		5929.2.a.k		2
77.l	even	10	1		5929.2.a.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.2.a.e	✓	2	11.c	even	5	1
847.2.a.g	yes	2	11.d	odd	10	1
847.2.f.o		8	11.b	odd	2	1
847.2.f.o		8	11.d	odd	10	3
847.2.f.r		8	1.a	even	1	1	trivial
847.2.f.r		8	11.c	even	5	3	inner
5929.2.a.k		2	77.j	odd	10	1
5929.2.a.p		2	77.l	even	10	1
7623.2.a.bc		2	33.f	even	10	1
7623.2.a.bs		2	33.h	odd	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{8} - T_{2}^{7} + 4T_{2}^{6} - 7T_{2}^{5} + 19T_{2}^{4} + 21T_{2}^{3} + 36T_{2}^{2} + 27T_{2} + 81 \)

\( T_{3}^{8} - T_{3}^{7} + 4T_{3}^{6} - 7T_{3}^{5} + 19T_{3}^{4} + 21T_{3}^{3} + 36T_{3}^{2} + 27T_{3} + 81 \)

\( T_{13}^{8} - 6T_{13}^{7} + 40T_{13}^{6} - 264T_{13}^{5} + 1744T_{13}^{4} + 1056T_{13}^{3} + 640T_{13}^{2} + 384T_{13} + 256 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - T^7 + 4*T^6 - 7*T^5 + 19*T^4 + 21*T^3 + 36*T^2 + 27*T + 81
$3$	T^8 - T^7 + 4*T^6 - 7*T^5 + 19*T^4 + 21*T^3 + 36*T^2 + 27*T + 81
$5$	T^8 + 13*T^6 + 169*T^4 + 2197*T^2 + 28561
$7$	(T^4 + T^3 + T^2 + T + 1)^2
$11$	\( T^{8} \)