Newform orbit 2646.2.d.d

• Label: 2646.2.d.d
• Level: $2646$
• Weight: $2$
• Character orbit: 2646.d
• 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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.1284163748\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	no (minimal twist has level 378)
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.

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

Basis of coefficient ring

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

Character values

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

\(n\)	\(785\)	\(1081\)
\(\chi(n)\)	\(-1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2645.1

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

−

1.00000i

0

−1.00000

−2.44949

0

0

1.00000i

0

2.44949i

2645.2

−

1.00000i

0

−1.00000

−2.44949

0

0

1.00000i

0

2.44949i

2645.3

−

1.00000i

0

−1.00000

2.44949

0

0

1.00000i

0

−

2.44949i

2645.4

−

1.00000i

0

−1.00000

2.44949

0

0

1.00000i

0

−

2.44949i

2645.5

1.00000i

0

−1.00000

−2.44949

0

0

−

1.00000i

0

−

2.44949i

2645.6

1.00000i

0

−1.00000

−2.44949

0

0

−

1.00000i

0

−

2.44949i

2645.7

1.00000i

0

−1.00000

2.44949

0

0

−

1.00000i

0

2.44949i

2645.8

1.00000i

0

−1.00000

2.44949

0

0

−

1.00000i

0

2.44949i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2646.2.d.d		8
3.b	odd	2	1	inner	2646.2.d.d		8
7.b	odd	2	1	inner	2646.2.d.d		8
7.c	even	3	1		378.2.k.d	✓	8
7.d	odd	6	1		378.2.k.d	✓	8
21.c	even	2	1	inner	2646.2.d.d		8
21.g	even	6	1		378.2.k.d	✓	8
21.h	odd	6	1		378.2.k.d	✓	8
63.g	even	3	1		1134.2.t.f		8
63.h	even	3	1		1134.2.l.e		8
63.i	even	6	1		1134.2.l.e		8
63.j	odd	6	1		1134.2.l.e		8
63.k	odd	6	1		1134.2.t.f		8
63.n	odd	6	1		1134.2.t.f		8
63.s	even	6	1		1134.2.t.f		8
63.t	odd	6	1		1134.2.l.e		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
378.2.k.d	✓	8	7.c	even	3	1
378.2.k.d	✓	8	7.d	odd	6	1
378.2.k.d	✓	8	21.g	even	6	1
378.2.k.d	✓	8	21.h	odd	6	1
1134.2.l.e		8	63.h	even	3	1
1134.2.l.e		8	63.i	even	6	1
1134.2.l.e		8	63.j	odd	6	1
1134.2.l.e		8	63.t	odd	6	1
1134.2.t.f		8	63.g	even	3	1
1134.2.t.f		8	63.k	odd	6	1
1134.2.t.f		8	63.n	odd	6	1
1134.2.t.f		8	63.s	even	6	1
2646.2.d.d		8	1.a	even	1	1	trivial
2646.2.d.d		8	3.b	odd	2	1	inner
2646.2.d.d		8	7.b	odd	2	1	inner
2646.2.d.d		8	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{4} \)
$3$	\( T^{8} \)
$5$	\( (T^{2} - 6)^{4} \)
$7$	\( T^{8} \)
$11$	\( (T^{2} + 18)^{4} \)