Newform orbit 2646.2.f.s

• Label: 2646.2.f.s
• Level: $2646$
• Weight: $2$
• Character orbit: 2646.f
• 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.f (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:	8.0.3317760000.3
Defining polynomial:	\( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} - 7\nu^{5} + 28\nu^{3} - 120\nu ) / 63 \)
\(\beta_{2}\)	\(=\)	\( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \)
\(\beta_{3}\)	\(=\)	\( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \)
\(\beta_{4}\)	\(=\)	\( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9 \)
\(\beta_{6}\)	\(=\)	\( ( -10\nu^{6} - 14\nu^{4} - 7\nu^{2} + 234 ) / 63 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} - \nu^{5} + 4\nu^{3} - 15\nu ) / 9 \)

Character values

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

\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(-1 + \beta_{3}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

883.1

−1.72286	−	0.178197i
−1.01575	−	1.40294i
1.01575	+	1.40294i
1.72286	+	0.178197i
−1.72286	+	0.178197i
−1.01575	+	1.40294i
1.01575	−	1.40294i
1.72286	−	0.178197i

0.500000

−

0.866025i

0

−0.500000

−

0.866025i

−1.72286

−

2.98408i

0

0

−1.00000

0

−3.44572

883.2

0.500000

−

0.866025i

0

−0.500000

−

0.866025i

−1.01575

−

1.75934i

0

0

−1.00000

0

−2.03151

883.3

0.500000

−

0.866025i

0

−0.500000

−

0.866025i

1.01575

+

1.75934i

0

0

−1.00000

0

2.03151

883.4

0.500000

−

0.866025i

0

−0.500000

−

0.866025i

1.72286

+

2.98408i

0

0

−1.00000

0

3.44572

1765.1

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

−1.72286

+

2.98408i

0

0

−1.00000

0

−3.44572

1765.2

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

−1.01575

+

1.75934i

0

0

−1.00000

0

−2.03151

1765.3

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

1.01575

−

1.75934i

0

0

−1.00000

0

2.03151

1765.4

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

1.72286

−

2.98408i

0

0

−1.00000

0

3.44572

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
9.c	even	3	1	inner
63.l	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.f.s		8
3.b	odd	2	1		882.2.f.p	✓	8
7.b	odd	2	1	inner	2646.2.f.s		8
7.c	even	3	1		2646.2.e.r		8
7.c	even	3	1		2646.2.h.s		8
7.d	odd	6	1		2646.2.e.r		8
7.d	odd	6	1		2646.2.h.s		8
9.c	even	3	1	inner	2646.2.f.s		8
9.c	even	3	1		7938.2.a.cd		4
9.d	odd	6	1		882.2.f.p	✓	8
9.d	odd	6	1		7938.2.a.cu		4
21.c	even	2	1		882.2.f.p	✓	8
21.g	even	6	1		882.2.e.t		8
21.g	even	6	1		882.2.h.r		8
21.h	odd	6	1		882.2.e.t		8
21.h	odd	6	1		882.2.h.r		8
63.g	even	3	1		2646.2.e.r		8
63.h	even	3	1		2646.2.h.s		8
63.i	even	6	1		882.2.h.r		8
63.j	odd	6	1		882.2.h.r		8
63.k	odd	6	1		2646.2.e.r		8
63.l	odd	6	1	inner	2646.2.f.s		8
63.l	odd	6	1		7938.2.a.cd		4
63.n	odd	6	1		882.2.e.t		8
63.o	even	6	1		882.2.f.p	✓	8
63.o	even	6	1		7938.2.a.cu		4
63.s	even	6	1		882.2.e.t		8
63.t	odd	6	1		2646.2.h.s		8

    

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

Hecke kernels

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

\( T_{5}^{8} + 16T_{5}^{6} + 207T_{5}^{4} + 784T_{5}^{2} + 2401 \)

\( T_{11}^{2} - 4T_{11} + 16 \)

\( T_{13}^{4} + 18T_{13}^{2} + 324 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{4} \)
$3$	\( T^{8} \)
$5$	T^8 + 16*T^6 + 207*T^4 + 784*T^2 + 2401
$7$	\( T^{8} \)
$11$	\( (T^{2} - 4 T + 16)^{4} \)