Newform orbit 8619.2.a.bg

• Label: 8619.2.a.bg
• Level: $8619$
• Weight: $2$
• Character orbit: 8619.a
• Self dual: yes
• Analytic conductor: $68.823$
• Analytic rank: $1$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(68.8230615021\)
Analytic rank:	\(1\)
Dimension:	\(8\)
Coefficient field:	8.8.59052888064.1
Defining polynomial:	\( x^{8} - 10x^{6} + 28x^{4} - 24x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 663)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{4} - \beta_1) q^{5} + \beta_1 q^{6} - \beta_{7} q^{7} + \beta_{3} q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 4 q^{4} + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{6} + 28x^{4} - 24x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 4\nu \)
\(\beta_{4}\)	\(=\)	\( \nu^{5} - 8\nu^{3} + 12\nu \)
\(\beta_{5}\)	\(=\)	\( \nu^{6} - 8\nu^{4} + 12\nu^{2} \)
\(\beta_{6}\)	\(=\)	\( -\nu^{6} + 9\nu^{4} - 19\nu^{2} + 7 \)
\(\beta_{7}\)	\(=\)	\( \nu^{7} - 9\nu^{5} + 20\nu^{3} - 12\nu \)

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

−2.44735
−1.53025
−1.27474
−0.209470
0.209470
1.27474
1.53025
2.44735

−2.44735

1.00000

3.98951

2.54977

−2.44735

−0.511026

−4.86902

1.00000

−6.24017

1.2

−1.53025

1.00000

0.341661

3.44293

−1.53025

−2.56617

2.53767

1.00000

−5.26854

1.3

−1.27474

1.00000

−0.375049

−0.816926

−1.27474

1.30719

3.02756

1.00000

1.04137

1.4

−0.209470

1.00000

−1.95612

−2.23105

−0.209470

−2.33343

0.828690

1.00000

0.467338

1.5

0.209470

1.00000

−1.95612

2.23105

0.209470

2.33343

−0.828690

1.00000

0.467338

1.6

1.27474

1.00000

−0.375049

0.816926

1.27474

−1.30719

−3.02756

1.00000

1.04137

1.7

1.53025

1.00000

0.341661

−3.44293

1.53025

2.56617

−2.53767

1.00000

−5.26854

1.8

2.44735

1.00000

3.98951

−2.54977

2.44735

0.511026

4.86902

1.00000

−6.24017

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(13\)	\(-1\)
\(17\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8619.2.a.bg		8
13.b	even	2	1	inner	8619.2.a.bg		8
13.d	odd	4	2		663.2.b.e	✓	8
39.f	even	4	2		1989.2.b.h		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
663.2.b.e	✓	8	13.d	odd	4	2
1989.2.b.h		8	39.f	even	4	2
8619.2.a.bg		8	1.a	even	1	1	trivial
8619.2.a.bg		8	13.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{8} - 10T_{2}^{6} + 28T_{2}^{4} - 24T_{2}^{2} + 1 \)

\( T_{5}^{8} - 24T_{5}^{6} + 184T_{5}^{4} - 496T_{5}^{2} + 256 \)

\( T_{7}^{8} - 14T_{7}^{6} + 60T_{7}^{4} - 76T_{7}^{2} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - 10*T^6 + 28*T^4 - 24*T^2 + 1
$3$	\( (T - 1)^{8} \)
$5$	T^8 - 24*T^6 + 184*T^4 - 496*T^2 + 256
$7$	T^8 - 14*T^6 + 60*T^4 - 76*T^2 + 16
$11$	T^8 - 52*T^6 + 824*T^4 - 4496*T^2 + 7744