Newform orbit 882.2.e.s

• Label: 882.2.e.s
• Level: $882$
• Weight: $2$
• Character orbit: 882.e
• Analytic conductor: $7.043$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.04280545828\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 + (-b7 + b6 + 2*b5 - b3) * q^3 + q^4 - 2*b7 * q^5 + (-b7 + b6 + 2*b5 - b3) * q^6 + q^8 + (b4 + b2) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8}+O(q^{10}) \)

Basis of coefficient ring

\(\beta_{1}\)	\(=\)	\( \zeta_{24}^{4} \)
\(\beta_{2}\)	\(=\)	\( \zeta_{24}^{6} + \zeta_{24}^{2} \)
\(\beta_{3}\)	\(=\)	\( \zeta_{24}^{7} + \zeta_{24} \)
\(\beta_{4}\)	\(=\)	\( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
\(\beta_{5}\)	\(=\)	\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
\(\beta_{6}\)	\(=\)	\( -\zeta_{24}^{7} + \zeta_{24}^{5} \)
\(\beta_{7}\)	\(=\)	\( -\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(785\)
\(\chi(n)\)	\(-\beta_{1}\)	\(-\beta_{1}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

373.1

−0.965926	+	0.258819i
0.258819	+	0.965926i
−0.258819	−	0.965926i
0.965926	−	0.258819i
−0.965926	−	0.258819i
0.258819	−	0.965926i
−0.258819	+	0.965926i
0.965926	+	0.258819i

1.00000

−1.67303

+

0.448288i

1.00000

1.93185

+

3.34607i

−1.67303

+

0.448288i

0

1.00000

2.59808

−

1.50000i

1.93185

+

3.34607i

373.2

1.00000

−0.448288

−

1.67303i

1.00000

−0.517638

−

0.896575i

−0.448288

−

1.67303i

0

1.00000

−2.59808

+

1.50000i

−0.517638

−

0.896575i

373.3

1.00000

0.448288

+

1.67303i

1.00000

0.517638

+

0.896575i

0.448288

+

1.67303i

0

1.00000

−2.59808

+

1.50000i

0.517638

+

0.896575i

373.4

1.00000

1.67303

−

0.448288i

1.00000

−1.93185

−

3.34607i

1.67303

−

0.448288i

0

1.00000

2.59808

−

1.50000i

−1.93185

−

3.34607i

655.1

1.00000

−1.67303

−

0.448288i

1.00000

1.93185

−

3.34607i

−1.67303

−

0.448288i

0

1.00000

2.59808

+

1.50000i

1.93185

−

3.34607i

655.2

1.00000

−0.448288

+

1.67303i

1.00000

−0.517638

+

0.896575i

−0.448288

+

1.67303i

0

1.00000

−2.59808

−

1.50000i

−0.517638

+

0.896575i

655.3

1.00000

0.448288

−

1.67303i

1.00000

0.517638

−

0.896575i

0.448288

−

1.67303i

0

1.00000

−2.59808

−

1.50000i

0.517638

−

0.896575i

655.4

1.00000

1.67303

+

0.448288i

1.00000

−1.93185

+

3.34607i

1.67303

+

0.448288i

0

1.00000

2.59808

+

1.50000i

−1.93185

+

3.34607i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
63.h	even	3	1	inner
63.t	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	882.2.e.s		8
3.b	odd	2	1		2646.2.e.q		8
7.b	odd	2	1	inner	882.2.e.s		8
7.c	even	3	1		882.2.f.q	✓	8
7.c	even	3	1		882.2.h.q		8
7.d	odd	6	1		882.2.f.q	✓	8
7.d	odd	6	1		882.2.h.q		8
9.c	even	3	1		882.2.h.q		8
9.d	odd	6	1		2646.2.h.t		8
21.c	even	2	1		2646.2.e.q		8
21.g	even	6	1		2646.2.f.r		8
21.g	even	6	1		2646.2.h.t		8
21.h	odd	6	1		2646.2.f.r		8
21.h	odd	6	1		2646.2.h.t		8
63.g	even	3	1		882.2.f.q	✓	8
63.h	even	3	1	inner	882.2.e.s		8
63.h	even	3	1		7938.2.a.cp		4
63.i	even	6	1		2646.2.e.q		8
63.i	even	6	1		7938.2.a.ci		4
63.j	odd	6	1		2646.2.e.q		8
63.j	odd	6	1		7938.2.a.ci		4
63.k	odd	6	1		882.2.f.q	✓	8
63.l	odd	6	1		882.2.h.q		8
63.n	odd	6	1		2646.2.f.r		8
63.o	even	6	1		2646.2.h.t		8
63.s	even	6	1		2646.2.f.r		8
63.t	odd	6	1	inner	882.2.e.s		8
63.t	odd	6	1		7938.2.a.cp		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
882.2.e.s		8	1.a	even	1	1	trivial
882.2.e.s		8	7.b	odd	2	1	inner
882.2.e.s		8	63.h	even	3	1	inner
882.2.e.s		8	63.t	odd	6	1	inner
882.2.f.q	✓	8	7.c	even	3	1
882.2.f.q	✓	8	7.d	odd	6	1
882.2.f.q	✓	8	63.g	even	3	1
882.2.f.q	✓	8	63.k	odd	6	1
882.2.h.q		8	7.c	even	3	1
882.2.h.q		8	7.d	odd	6	1
882.2.h.q		8	9.c	even	3	1
882.2.h.q		8	63.l	odd	6	1
2646.2.e.q		8	3.b	odd	2	1
2646.2.e.q		8	21.c	even	2	1
2646.2.e.q		8	63.i	even	6	1
2646.2.e.q		8	63.j	odd	6	1
2646.2.f.r		8	21.g	even	6	1
2646.2.f.r		8	21.h	odd	6	1
2646.2.f.r		8	63.n	odd	6	1
2646.2.f.r		8	63.s	even	6	1
2646.2.h.t		8	9.d	odd	6	1
2646.2.h.t		8	21.g	even	6	1
2646.2.h.t		8	21.h	odd	6	1
2646.2.h.t		8	63.o	even	6	1
7938.2.a.ci		4	63.i	even	6	1
7938.2.a.ci		4	63.j	odd	6	1
7938.2.a.cp		4	63.h	even	3	1
7938.2.a.cp		4	63.t	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} + 16T_{5}^{6} + 240T_{5}^{4} + 256T_{5}^{2} + 256 \)

\( T_{11}^{4} - 4T_{11}^{3} + 15T_{11}^{2} - 4T_{11} + 1 \)

\( T_{13}^{8} + 48T_{13}^{6} + 2160T_{13}^{4} + 6912T_{13}^{2} + 20736 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{8} \)
$3$	\( T^{8} - 9T^{4} + 81 \)
$5$	T^8 + 16*T^6 + 240*T^4 + 256*T^2 + 256
$7$	\( T^{8} \)
$11$	(T^4 - 4*T^3 + 15*T^2 - 4*T + 1)^2