Newform orbit 4598.2.a.bw

• Label: 4598.2.a.bw
• Level: $4598$
• Weight: $2$
• Character orbit: 4598.a
• Self dual: yes
• Analytic conductor: $36.715$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(36.7152148494\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - 2x^{7} - 11x^{6} + 22x^{5} + 34x^{4} - 68x^{3} - 28x^{2} + 60x - 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 418)
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.

\(f(q)\)	\(=\)	\( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{6} - \beta_{3} - \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} + 2) q^{7} - q^{8} + (\beta_{6} - \beta_{5} + \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} + 2 q^{6} + 14 q^{7} - 8 q^{8} + 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 11x^{6} + 22x^{5} + 34x^{4} - 68x^{3} - 28x^{2} + 60x - 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{5} + 9\nu^{3} - 18\nu ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} - 9\nu^{5} + 2\nu^{4} + 20\nu^{3} - 10\nu^{2} - 12\nu + 8 ) / 4 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} - 11\nu^{5} + 36\nu^{3} + 2\nu^{2} - 36\nu ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 11\nu^{5} - \nu^{4} - 35\nu^{3} + 4\nu^{2} + 32\nu - 2 ) / 2 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{7} + 11\nu^{5} - \nu^{4} - 35\nu^{3} + 6\nu^{2} + 30\nu - 8 ) / 2 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + 11\nu^{5} - 9\nu^{4} - 34\nu^{3} + 16\nu^{2} + 28\nu ) / 2 \)

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

2.40829
2.27943
1.59897
1.18171
0.0692689
−1.25705
−1.75021
−2.53041

−1.00000

−2.40829

1.00000

−3.43347

2.40829

1.05667

−1.00000

2.79988

3.43347

1.2

−1.00000

−2.27943

1.00000

−2.24748

2.27943

4.63064

−1.00000

2.19578

2.24748

1.3

−1.00000

−1.59897

1.00000

3.74775

1.59897

−0.838369

−1.00000

−0.443297

−3.74775

1.4

−1.00000

−1.18171

1.00000

−0.117838

1.18171

−1.74369

−1.00000

−1.60355

0.117838

1.5

−1.00000

−0.0692689

1.00000

1.10125

0.0692689

1.99611

−1.00000

−2.99520

−1.10125

1.6

−1.00000

1.25705

1.00000

1.68320

−1.25705

4.32620

−1.00000

−1.41983

−1.68320

1.7

−1.00000

1.75021

1.00000

−2.99749

−1.75021

0.219425

−1.00000

0.0632464

2.99749

1.8

−1.00000

2.53041

1.00000

0.264060

−2.53041

4.35301

−1.00000

3.40297

−0.264060

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(11\)	\(1\)
\(19\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4598.2.a.bw		8
11.b	odd	2	1		4598.2.a.bz		8
11.c	even	5	2		418.2.f.g	✓	16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
418.2.f.g	✓	16	11.c	even	5	2
4598.2.a.bw		8	1.a	even	1	1	trivial
4598.2.a.bz		8	11.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{8} + 2T_{3}^{7} - 11T_{3}^{6} - 22T_{3}^{5} + 34T_{3}^{4} + 68T_{3}^{3} - 28T_{3}^{2} - 60T_{3} - 4 \)

\( T_{5}^{8} + 2T_{5}^{7} - 20T_{5}^{6} - 36T_{5}^{5} + 99T_{5}^{4} + 100T_{5}^{3} - 180T_{5}^{2} + 20T_{5} + 5 \)

\( T_{7}^{8} - 14T_{7}^{7} + 64T_{7}^{6} - 70T_{7}^{5} - 211T_{7}^{4} + 442T_{7}^{3} + 10T_{7}^{2} - 290T_{7} + 59 \)

\( T_{13}^{8} - 9T_{13}^{7} - 23T_{13}^{6} + 394T_{13}^{5} - 744T_{13}^{4} - 1240T_{13}^{3} + 2080T_{13}^{2} + 1760T_{13} + 320 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{8} \)
$3$	T^8 + 2*T^7 - 11*T^6 - 22*T^5 + 34*T^4 + 68*T^3 - 28*T^2 - 60*T - 4
$5$	T^8 + 2*T^7 - 20*T^6 - 36*T^5 + 99*T^4 + 100*T^3 - 180*T^2 + 20*T + 5
$7$	T^8 - 14*T^7 + 64*T^6 - 70*T^5 - 211*T^4 + 442*T^3 + 10*T^2 - 290*T + 59
$11$	\( T^{8} \)