Newform orbit 588.2.e.b

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

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.69520363885\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.3317760000.1
Defining polynomial:	\( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q - \beta_{6} q^{2} - \beta_{2} q^{3} + ( - \beta_{3} - 1) q^{4} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{5} + (\beta_{4} - \beta_1) q^{6} + (\beta_{6} - 2 \beta_{5}) q^{8} + (\beta_{6} + \beta_{5} + 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} + 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} + 14\nu^{5} - 49\nu^{3} - 6\nu ) / 48 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{4} - 4\nu^{2} - 3 ) / 3 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} - 6\nu^{5} + 17\nu^{3} - 18\nu ) / 24 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{6} + 6\nu^{4} - 17\nu^{2} + 18 ) / 24 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{6} + 6\nu^{4} - 5\nu^{2} - 6 ) / 12 \)
\(\beta_{7}\)	\(=\)	\( ( -3\nu^{7} + 26\nu^{5} - 35\nu^{3} - 114\nu ) / 48 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

491.1

−2.15988	+	0.258819i
2.15988	−	0.258819i
−2.15988	−	0.258819i
2.15988	+	0.258819i
0.578737	−	0.965926i
−0.578737	+	0.965926i
0.578737	+	0.965926i
−0.578737	−	0.965926i

−0.866025

−

1.11803i

−1.58114

−

0.707107i

−0.500000

+

1.93649i

−

2.44949i

0.578737

+

2.38014i

0

2.59808

−

1.11803i

2.00000

+

2.23607i

−2.73861

+

2.12132i

491.2

−0.866025

−

1.11803i

1.58114

+

0.707107i

−0.500000

+

1.93649i

2.44949i

−0.578737

−

2.38014i

0

2.59808

−

1.11803i

2.00000

+

2.23607i

2.73861

−

2.12132i

491.3

−0.866025

+

1.11803i

−1.58114

+

0.707107i

−0.500000

−

1.93649i

2.44949i

0.578737

−

2.38014i

0

2.59808

+

1.11803i

2.00000

−

2.23607i

−2.73861

−

2.12132i

491.4

−0.866025

+

1.11803i

1.58114

−

0.707107i

−0.500000

−

1.93649i

−

2.44949i

−0.578737

+

2.38014i

0

2.59808

+

1.11803i

2.00000

−

2.23607i

2.73861

+

2.12132i

491.5

0.866025

−

1.11803i

−1.58114

−

0.707107i

−0.500000

−

1.93649i

2.44949i

−2.15988

+

1.15539i

0

−2.59808

−

1.11803i

2.00000

+

2.23607i

2.73861

+

2.12132i

491.6

0.866025

−

1.11803i

1.58114

+

0.707107i

−0.500000

−

1.93649i

−

2.44949i

2.15988

−

1.15539i

0

−2.59808

−

1.11803i

2.00000

+

2.23607i

−2.73861

−

2.12132i

491.7

0.866025

+

1.11803i

−1.58114

+

0.707107i

−0.500000

+

1.93649i

−

2.44949i

−2.15988

−

1.15539i

0

−2.59808

+

1.11803i

2.00000

−

2.23607i

2.73861

−

2.12132i

491.8

0.866025

+

1.11803i

1.58114

−

0.707107i

−0.500000

+

1.93649i

2.44949i

2.15988

+

1.15539i

0

−2.59808

+

1.11803i

2.00000

−

2.23607i

−2.73861

+

2.12132i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.b	odd	2	1	inner
12.b	even	2	1	inner
21.c	even	2	1	inner
28.d	even	2	1	inner
84.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.2.e.b	✓	8
3.b	odd	2	1	inner	588.2.e.b	✓	8
4.b	odd	2	1	inner	588.2.e.b	✓	8
7.b	odd	2	1	inner	588.2.e.b	✓	8
7.c	even	3	2		588.2.n.c		16
7.d	odd	6	2		588.2.n.c		16
12.b	even	2	1	inner	588.2.e.b	✓	8
21.c	even	2	1	inner	588.2.e.b	✓	8
21.g	even	6	2		588.2.n.c		16
21.h	odd	6	2		588.2.n.c		16
28.d	even	2	1	inner	588.2.e.b	✓	8
28.f	even	6	2		588.2.n.c		16
28.g	odd	6	2		588.2.n.c		16
84.h	odd	2	1	inner	588.2.e.b	✓	8
84.j	odd	6	2		588.2.n.c		16
84.n	even	6	2		588.2.n.c		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
588.2.e.b	✓	8	1.a	even	1	1	trivial
588.2.e.b	✓	8	3.b	odd	2	1	inner
588.2.e.b	✓	8	4.b	odd	2	1	inner
588.2.e.b	✓	8	7.b	odd	2	1	inner
588.2.e.b	✓	8	12.b	even	2	1	inner
588.2.e.b	✓	8	21.c	even	2	1	inner
588.2.e.b	✓	8	28.d	even	2	1	inner
588.2.e.b	✓	8	84.h	odd	2	1	inner
588.2.n.c		16	7.c	even	3	2
588.2.n.c		16	7.d	odd	6	2
588.2.n.c		16	21.g	even	6	2
588.2.n.c		16	21.h	odd	6	2
588.2.n.c		16	28.f	even	6	2
588.2.n.c		16	28.g	odd	6	2
588.2.n.c		16	84.j	odd	6	2
588.2.n.c		16	84.n	even	6	2

Hecke kernels

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

\( T_{5}^{2} + 6 \)

\( T_{11}^{2} - 12 \)

\( T_{13}^{2} - 30 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} + T^{2} + 4)^{2} \)
$3$	\( (T^{4} - 4 T^{2} + 9)^{2} \)
$5$	\( (T^{2} + 6)^{4} \)
$7$	\( T^{8} \)
$11$	\( (T^{2} - 12)^{4} \)