Newform orbit 392.2.p.c

• Label: 392.2.p.c
• Level: $392$
• Weight: $2$
• Character orbit: 392.p
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -7
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial:	\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) :

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

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

165.1

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

−0.895644

+

1.09445i

0

−0.395644

−

1.96048i

0

0

0

2.50000

+

1.32288i

−1.50000

−

2.59808i

0

165.2

1.39564

−

0.228425i

0

1.89564

−

0.637600i

0

0

0

2.50000

−

1.32288i

−1.50000

−

2.59808i

0

373.1

−0.895644

−

1.09445i

0

−0.395644

+

1.96048i

0

0

0

2.50000

−

1.32288i

−1.50000

+

2.59808i

0

373.2

1.39564

+

0.228425i

0

1.89564

+

0.637600i

0

0

0

2.50000

+

1.32288i

−1.50000

+

2.59808i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
7.c	even	3	1	inner
7.d	odd	6	1	inner
8.b	even	2	1	inner
56.h	odd	2	1	inner
56.j	odd	6	1	inner
56.p	even	6	1	inner

Twists

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

    

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

Hecke kernels

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

\( T_{3} \)

\( T_{17} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 - T^3 - T^2 - 2*T + 4
$3$	\( T^{4} \)
$5$	\( T^{4} \)
$7$	\( T^{4} \)
$11$	\( T^{4} - 28T^{2} + 784 \)