Newform orbit 392.2.e.b

• Label: 392.2.e.b
• Level: $392$
• Weight: $2$
• Character orbit: 392.e
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	4.0.2048.2
Defining polynomial:	\( x^{4} + 4x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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.

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

195.1

	−	1.84776i
		1.84776i
	−	0.765367i
		0.765367i

−1.41421

−

1.84776i

2.00000

0

2.61313i

0

−2.82843

−0.414214

0

195.2

−1.41421

1.84776i

2.00000

0

−

2.61313i

0

−2.82843

−0.414214

0

195.3

1.41421

−

0.765367i

2.00000

0

−

1.08239i

0

2.82843

2.41421

0

195.4

1.41421

0.765367i

2.00000

0

1.08239i

0

2.82843

2.41421

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
7.b	odd	2	1	inner
56.e	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.2.e.b	✓	4
4.b	odd	2	1		1568.2.e.b		4
7.b	odd	2	1	inner	392.2.e.b	✓	4
7.c	even	3	2		392.2.m.b		8
7.d	odd	6	2		392.2.m.b		8
8.b	even	2	1		1568.2.e.b		4
8.d	odd	2	1	CM	392.2.e.b	✓	4
28.d	even	2	1		1568.2.e.b		4
28.f	even	6	2		1568.2.q.b		8
28.g	odd	6	2		1568.2.q.b		8
56.e	even	2	1	inner	392.2.e.b	✓	4
56.h	odd	2	1		1568.2.e.b		4
56.j	odd	6	2		1568.2.q.b		8
56.k	odd	6	2		392.2.m.b		8
56.m	even	6	2		392.2.m.b		8
56.p	even	6	2		1568.2.q.b		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.2.e.b	✓	4	1.a	even	1	1	trivial
392.2.e.b	✓	4	7.b	odd	2	1	inner
392.2.e.b	✓	4	8.d	odd	2	1	CM
392.2.e.b	✓	4	56.e	even	2	1	inner
392.2.m.b		8	7.c	even	3	2
392.2.m.b		8	7.d	odd	6	2
392.2.m.b		8	56.k	odd	6	2
392.2.m.b		8	56.m	even	6	2
1568.2.e.b		4	4.b	odd	2	1
1568.2.e.b		4	8.b	even	2	1
1568.2.e.b		4	28.d	even	2	1
1568.2.e.b		4	56.h	odd	2	1
1568.2.q.b		8	28.f	even	6	2
1568.2.q.b		8	28.g	odd	6	2
1568.2.q.b		8	56.j	odd	6	2
1568.2.q.b		8	56.p	even	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):

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

\( T_{5} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{2} \)
$3$	\( T^{4} + 4T^{2} + 2 \)
$5$	\( T^{4} \)
$7$	\( T^{4} \)
$11$	\( (T^{2} - 4 T - 14)^{2} \)