Newform orbit 2646.2.h.r

• Label: 2646.2.h.r
• Level: $2646$
• Weight: $2$
• Character orbit: 2646.h
• Analytic conductor: $21.128$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.1284163748\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.31116960000.2
Defining polynomial:	\( x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 882)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} + 8\nu^{5} - 64\nu^{3} + 135\nu ) / 216 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{6} + 8\nu^{4} + 8\nu^{2} - 81 ) / 72 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} + 8\nu^{5} + 44\nu^{3} + 27\nu ) / 108 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} + 4\nu^{5} + 4\nu^{3} - 15\nu ) / 36 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{6} + \nu^{4} - 17\nu^{2} ) / 9 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} + 8\nu^{2} + 17 ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} + 2\nu^{5} + 2\nu^{3} - 15\nu ) / 18 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(-1 - \beta_{2}\)	\(\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

361.1

1.62968	−	0.586627i
0.306808	+	1.70466i
−0.306808	−	1.70466i
−1.62968	+	0.586627i
1.62968	+	0.586627i
0.306808	−	1.70466i
−0.306808	+	1.70466i
−1.62968	−	0.586627i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−3.25937

0

0

1.00000

0

1.62968

−

2.82269i

361.2

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−0.613616

0

0

1.00000

0

0.306808

−

0.531407i

361.3

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

0.613616

0

0

1.00000

0

−0.306808

+

0.531407i

361.4

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

3.25937

0

0

1.00000

0

−1.62968

+

2.82269i

667.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−3.25937

0

0

1.00000

0

1.62968

+

2.82269i

667.2

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−0.613616

0

0

1.00000

0

0.306808

+

0.531407i

667.3

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

0.613616

0

0

1.00000

0

−0.306808

−

0.531407i

667.4

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

3.25937

0

0

1.00000

0

−1.62968

−

2.82269i

Inner twists

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

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\):

\( T_{5}^{4} - 11T_{5}^{2} + 4 \)

\( T_{11}^{2} + T_{11} - 26 \)

\( T_{13}^{8} + 44T_{13}^{6} + 1872T_{13}^{4} + 2816T_{13}^{2} + 4096 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{4} \)
$3$	\( T^{8} \)
$5$	\( (T^{4} - 11 T^{2} + 4)^{2} \)
$7$	\( T^{8} \)
$11$	\( (T^{2} + T - 26)^{4} \)