Newform orbit 588.2.b.b

• Label: 588.2.b.b
• Level: $588$
• Weight: $2$
• Character orbit: 588.b
• Analytic conductor: $4.695$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.69520363885\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.562828176.1
Defining polynomial:	\( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 84)
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.

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

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

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} + \nu^{5} + 4\nu^{4} + 2\nu^{3} + 4\nu - 16 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} + \nu^{5} + 4\nu^{4} - 6\nu^{3} + 4\nu - 16 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + \nu^{5} - 3\nu^{4} + 4\nu^{3} + 2\nu^{2} - 8\nu + 12 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 2\nu^{4} - 6\nu^{3} + 4\nu^{2} + 4\nu - 12 ) / 4 \)
\(\beta_{7}\)	\(=\)	\( ( -2\nu^{7} + \nu^{6} - 3\nu^{4} + 6\nu^{3} + 2\nu^{2} - 8\nu + 20 ) / 4 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(493\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

391.1

1.40376	+	0.171630i
1.40376	−	0.171630i
0.856419	+	1.12541i
0.856419	−	1.12541i
0.0777157	+	1.41208i
0.0777157	−	1.41208i
−1.33790	+	0.458297i
−1.33790	−	0.458297i

−1.40376

−

0.171630i

1.00000

1.94109

+

0.481855i

−

0.963711i

−1.40376

−

0.171630i

0

−2.64212

−

1.00956i

1.00000

−0.165402

+

1.35282i

391.2

−1.40376

+

0.171630i

1.00000

1.94109

−

0.481855i

0.963711i

−1.40376

+

0.171630i

0

−2.64212

+

1.00956i

1.00000

−0.165402

−

1.35282i

391.3

−0.856419

−

1.12541i

1.00000

−0.533092

+

1.92764i

−

3.85529i

−0.856419

−

1.12541i

0

2.62594

−

1.05092i

1.00000

−4.33878

+

3.30174i

391.4

−0.856419

+

1.12541i

1.00000

−0.533092

−

1.92764i

3.85529i

−0.856419

+

1.12541i

0

2.62594

+

1.05092i

1.00000

−4.33878

−

3.30174i

391.5

−0.0777157

−

1.41208i

1.00000

−1.98792

+

0.219481i

−

0.438962i

−0.0777157

−

1.41208i

0

0.464416

+

2.79004i

1.00000

−0.619848

+

0.0341142i

391.6

−0.0777157

+

1.41208i

1.00000

−1.98792

−

0.219481i

0.438962i

−0.0777157

+

1.41208i

0

0.464416

−

2.79004i

1.00000

−0.619848

−

0.0341142i

391.7

1.33790

−

0.458297i

1.00000

1.57993

−

1.22631i

2.45262i

1.33790

−

0.458297i

0

1.55176

−

2.36475i

1.00000

1.12403

+

3.28134i

391.8

1.33790

+

0.458297i

1.00000

1.57993

+

1.22631i

−

2.45262i

1.33790

+

0.458297i

0

1.55176

+

2.36475i

1.00000

1.12403

−

3.28134i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.2.b.b		8
3.b	odd	2	1		1764.2.b.i		8
4.b	odd	2	1		588.2.b.a		8
7.b	odd	2	1		588.2.b.a		8
7.c	even	3	1		84.2.o.a	✓	8
7.c	even	3	1		588.2.o.b		8
7.d	odd	6	1		84.2.o.b	yes	8
7.d	odd	6	1		588.2.o.d		8
12.b	even	2	1		1764.2.b.j		8
21.c	even	2	1		1764.2.b.j		8
21.g	even	6	1		252.2.bf.f		8
21.h	odd	6	1		252.2.bf.g		8
28.d	even	2	1	inner	588.2.b.b		8
28.f	even	6	1		84.2.o.a	✓	8
28.f	even	6	1		588.2.o.b		8
28.g	odd	6	1		84.2.o.b	yes	8
28.g	odd	6	1		588.2.o.d		8
56.j	odd	6	1		1344.2.bl.i		8
56.k	odd	6	1		1344.2.bl.i		8
56.m	even	6	1		1344.2.bl.j		8
56.p	even	6	1		1344.2.bl.j		8
84.h	odd	2	1		1764.2.b.i		8
84.j	odd	6	1		252.2.bf.g		8
84.n	even	6	1		252.2.bf.f		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.2.o.a	✓	8	7.c	even	3	1
84.2.o.a	✓	8	28.f	even	6	1
84.2.o.b	yes	8	7.d	odd	6	1
84.2.o.b	yes	8	28.g	odd	6	1
252.2.bf.f		8	21.g	even	6	1
252.2.bf.f		8	84.n	even	6	1
252.2.bf.g		8	21.h	odd	6	1
252.2.bf.g		8	84.j	odd	6	1
588.2.b.a		8	4.b	odd	2	1
588.2.b.a		8	7.b	odd	2	1
588.2.b.b		8	1.a	even	1	1	trivial
588.2.b.b		8	28.d	even	2	1	inner
588.2.o.b		8	7.c	even	3	1
588.2.o.b		8	28.f	even	6	1
588.2.o.d		8	7.d	odd	6	1
588.2.o.d		8	28.g	odd	6	1
1344.2.bl.i		8	56.j	odd	6	1
1344.2.bl.i		8	56.k	odd	6	1
1344.2.bl.j		8	56.m	even	6	1
1344.2.bl.j		8	56.p	even	6	1
1764.2.b.i		8	3.b	odd	2	1
1764.2.b.i		8	84.h	odd	2	1
1764.2.b.j		8	12.b	even	2	1
1764.2.b.j		8	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{8} + 22T_{5}^{6} + 113T_{5}^{4} + 104T_{5}^{2} + 16 \)

\( T_{19}^{4} - 6T_{19}^{3} - 7T_{19}^{2} + 60T_{19} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + 2*T^7 + T^6 - 2*T^5 - 6*T^4 - 4*T^3 + 4*T^2 + 16*T + 16
$3$	\( (T - 1)^{8} \)
$5$	T^8 + 22*T^6 + 113*T^4 + 104*T^2 + 16
$7$	\( T^{8} \)
$11$	T^8 + 38*T^6 + 257*T^4 + 568*T^2 + 400