Newform orbit 1078.2.e.r

• Label: 1078.2.e.r
• Level: $1078$
• Weight: $2$
• Character orbit: 1078.e
• Analytic conductor: $8.608$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.60787333789\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} + 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{4} - \beta_{3} q^{6} - q^{8} + 5 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 10 q^{9}+O(q^{10}) \)

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

\(\beta_{1}\)	\(=\)	\( 2\nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(981\)
\(\chi(n)\)	\(\beta_{2}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

−0.707107	+	1.22474i
0.707107	−	1.22474i
−0.707107	−	1.22474i
0.707107	+	1.22474i

0.500000

+

0.866025i

−1.41421

+

2.44949i

−0.500000

+

0.866025i

0

−2.82843

0

−1.00000

−2.50000

−

4.33013i

0

67.2

0.500000

+

0.866025i

1.41421

−

2.44949i

−0.500000

+

0.866025i

0

2.82843

0

−1.00000

−2.50000

−

4.33013i

0

177.1

0.500000

−

0.866025i

−1.41421

−

2.44949i

−0.500000

−

0.866025i

0

−2.82843

0

−1.00000

−2.50000

+

4.33013i

0

177.2

0.500000

−

0.866025i

1.41421

+

2.44949i

−0.500000

−

0.866025i

0

2.82843

0

−1.00000

−2.50000

+

4.33013i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1078.2.e.r		4
7.b	odd	2	1	inner	1078.2.e.r		4
7.c	even	3	1		1078.2.a.r	✓	2
7.c	even	3	1	inner	1078.2.e.r		4
7.d	odd	6	1		1078.2.a.r	✓	2
7.d	odd	6	1	inner	1078.2.e.r		4
21.g	even	6	1		9702.2.a.dn		2
21.h	odd	6	1		9702.2.a.dn		2
28.f	even	6	1		8624.2.a.by		2
28.g	odd	6	1		8624.2.a.by		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1078.2.a.r	✓	2	7.c	even	3	1
1078.2.a.r	✓	2	7.d	odd	6	1
1078.2.e.r		4	1.a	even	1	1	trivial
1078.2.e.r		4	7.b	odd	2	1	inner
1078.2.e.r		4	7.c	even	3	1	inner
1078.2.e.r		4	7.d	odd	6	1	inner
8624.2.a.by		2	28.f	even	6	1
8624.2.a.by		2	28.g	odd	6	1
9702.2.a.dn		2	21.g	even	6	1
9702.2.a.dn		2	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 64 \)

\( T_{5} \)

\( T_{13}^{2} - 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{2} \)
$3$	\( T^{4} + 8T^{2} + 64 \)
$5$	\( T^{4} \)
$7$	\( T^{4} \)
$11$	\( (T^{2} - T + 1)^{2} \)