Newform orbit 1280.2.n.n

• Label: 1280.2.n.n
• Level: $1280$
• Weight: $2$
• Character orbit: 1280.n
• Analytic conductor: $10.221$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.2208514587\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	8.0.49787136.1
Defining polynomial:	\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{6} \)
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - \nu^{5} + \nu^{3} + 8\nu ) / 8 \)
\(\beta_{3}\)	\(=\)	\( ( -5\nu^{7} - 2\nu^{6} - 15\nu^{5} + 10\nu^{4} - 25\nu^{3} + 30\nu^{2} - 20\nu + 16 ) / 40 \)
\(\beta_{4}\)	\(=\)	\( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 26 ) / 10 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} - 2\nu^{6} + 3\nu^{5} - 6\nu^{4} + 5\nu^{3} - 2\nu^{2} + 20\nu - 16 ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} + 2\nu^{6} + 3\nu^{5} + 6\nu^{4} + 5\nu^{3} + 2\nu^{2} + 20\nu + 16 ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( -5\nu^{7} + \nu^{6} - 5\nu^{5} - 5\nu^{4} - 15\nu^{3} - 15\nu^{2} - 30\nu - 8 ) / 20 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\)	\(257\)	\(261\)	\(511\)
\(\chi(n)\)	\(-\beta_{1}\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

767.1

1.09445	−	0.895644i
0.228425	+	1.39564i
−0.228425	+	1.39564i
−1.09445	−	0.895644i
1.09445	+	0.895644i
0.228425	−	1.39564i
−0.228425	−	1.39564i
−1.09445	+	0.895644i

0

−2.18890

−

2.18890i

0

2.18890

+

0.456850i

0

1.79129

−

1.79129i

0

6.58258i

0

767.2

0

−0.456850

−

0.456850i

0

0.456850

+

2.18890i

0

−2.79129

+

2.79129i

0

−

2.58258i

0

767.3

0

0.456850

+

0.456850i

0

−0.456850

−

2.18890i

0

−2.79129

+

2.79129i

0

−

2.58258i

0

767.4

0

2.18890

+

2.18890i

0

−2.18890

−

0.456850i

0

1.79129

−

1.79129i

0

6.58258i

0

1023.1

0

−2.18890

+

2.18890i

0

2.18890

−

0.456850i

0

1.79129

+

1.79129i

0

−

6.58258i

0

1023.2

0

−0.456850

+

0.456850i

0

0.456850

−

2.18890i

0

−2.79129

−

2.79129i

0

2.58258i

0

1023.3

0

0.456850

−

0.456850i

0

−0.456850

+

2.18890i

0

−2.79129

−

2.79129i

0

2.58258i

0

1023.4

0

2.18890

−

2.18890i

0

−2.18890

+

0.456850i

0

1.79129

+

1.79129i

0

−

6.58258i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
20.e	even	4	1	inner
40.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1280.2.n.n		8
4.b	odd	2	1		1280.2.n.p		8
5.c	odd	4	1		1280.2.n.p		8
8.b	even	2	1	inner	1280.2.n.n		8
8.d	odd	2	1		1280.2.n.p		8
16.e	even	4	2		320.2.o.f	yes	8
16.f	odd	4	2		320.2.o.e	✓	8
20.e	even	4	1	inner	1280.2.n.n		8
40.i	odd	4	1		1280.2.n.p		8
40.k	even	4	1	inner	1280.2.n.n		8
80.i	odd	4	1		320.2.o.e	✓	8
80.i	odd	4	1		1600.2.o.l		8
80.j	even	4	1		320.2.o.f	yes	8
80.j	even	4	1		1600.2.o.e		8
80.k	odd	4	2		1600.2.o.l		8
80.q	even	4	2		1600.2.o.e		8
80.s	even	4	1		320.2.o.f	yes	8
80.s	even	4	1		1600.2.o.e		8
80.t	odd	4	1		320.2.o.e	✓	8
80.t	odd	4	1		1600.2.o.l		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
320.2.o.e	✓	8	16.f	odd	4	2
320.2.o.e	✓	8	80.i	odd	4	1
320.2.o.e	✓	8	80.t	odd	4	1
320.2.o.f	yes	8	16.e	even	4	2
320.2.o.f	yes	8	80.j	even	4	1
320.2.o.f	yes	8	80.s	even	4	1
1280.2.n.n		8	1.a	even	1	1	trivial
1280.2.n.n		8	8.b	even	2	1	inner
1280.2.n.n		8	20.e	even	4	1	inner
1280.2.n.n		8	40.k	even	4	1	inner
1280.2.n.p		8	4.b	odd	2	1
1280.2.n.p		8	5.c	odd	4	1
1280.2.n.p		8	8.d	odd	2	1
1280.2.n.p		8	40.i	odd	4	1
1600.2.o.e		8	80.j	even	4	1
1600.2.o.e		8	80.q	even	4	2
1600.2.o.e		8	80.s	even	4	1
1600.2.o.l		8	80.i	odd	4	1
1600.2.o.l		8	80.k	odd	4	2
1600.2.o.l		8	80.t	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{8} + 92T_{3}^{4} + 16 \)

\( T_{7}^{4} + 2T_{7}^{3} + 2T_{7}^{2} - 20T_{7} + 100 \)

\( T_{13}^{4} + 36 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	\( T^{8} + 92T^{4} + 16 \)
$5$	\( T^{8} - 34T^{4} + 625 \)
$7$	(T^4 + 2*T^3 + 2*T^2 - 20*T + 100)^2
$11$	\( (T^{4} + 20 T^{2} + 16)^{2} \)