Newform orbit 392.2.i.h

• Label: 392.2.i.h
• Level: $392$
• Weight: $2$
• Character orbit: 392.i
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
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:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
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.

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

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

177.1

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

0

−0.707107

−

1.22474i

0

−1.41421

+

2.44949i

0

0

0

0.500000

−

0.866025i

0

177.2

0

0.707107

+

1.22474i

0

1.41421

−

2.44949i

0

0

0

0.500000

−

0.866025i

0

361.1

0

−0.707107

+

1.22474i

0

−1.41421

−

2.44949i

0

0

0

0.500000

+

0.866025i

0

361.2

0

0.707107

−

1.22474i

0

1.41421

+

2.44949i

0

0

0

0.500000

+

0.866025i

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	392.2.i.h		4
3.b	odd	2	1		3528.2.s.bj		4
4.b	odd	2	1		784.2.i.n		4
7.b	odd	2	1	inner	392.2.i.h		4
7.c	even	3	1		392.2.a.g	✓	2
7.c	even	3	1	inner	392.2.i.h		4
7.d	odd	6	1		392.2.a.g	✓	2
7.d	odd	6	1	inner	392.2.i.h		4
21.c	even	2	1		3528.2.s.bj		4
21.g	even	6	1		3528.2.a.be		2
21.g	even	6	1		3528.2.s.bj		4
21.h	odd	6	1		3528.2.a.be		2
21.h	odd	6	1		3528.2.s.bj		4
28.d	even	2	1		784.2.i.n		4
28.f	even	6	1		784.2.a.k		2
28.f	even	6	1		784.2.i.n		4
28.g	odd	6	1		784.2.a.k		2
28.g	odd	6	1		784.2.i.n		4
35.i	odd	6	1		9800.2.a.bv		2
35.j	even	6	1		9800.2.a.bv		2
56.j	odd	6	1		3136.2.a.bk		2
56.k	odd	6	1		3136.2.a.bp		2
56.m	even	6	1		3136.2.a.bp		2
56.p	even	6	1		3136.2.a.bk		2
84.j	odd	6	1		7056.2.a.ct		2
84.n	even	6	1		7056.2.a.ct		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.2.a.g	✓	2	7.c	even	3	1
392.2.a.g	✓	2	7.d	odd	6	1
392.2.i.h		4	1.a	even	1	1	trivial
392.2.i.h		4	7.b	odd	2	1	inner
392.2.i.h		4	7.c	even	3	1	inner
392.2.i.h		4	7.d	odd	6	1	inner
784.2.a.k		2	28.f	even	6	1
784.2.a.k		2	28.g	odd	6	1
784.2.i.n		4	4.b	odd	2	1
784.2.i.n		4	28.d	even	2	1
784.2.i.n		4	28.f	even	6	1
784.2.i.n		4	28.g	odd	6	1
3136.2.a.bk		2	56.j	odd	6	1
3136.2.a.bk		2	56.p	even	6	1
3136.2.a.bp		2	56.k	odd	6	1
3136.2.a.bp		2	56.m	even	6	1
3528.2.a.be		2	21.g	even	6	1
3528.2.a.be		2	21.h	odd	6	1
3528.2.s.bj		4	3.b	odd	2	1
3528.2.s.bj		4	21.c	even	2	1
3528.2.s.bj		4	21.g	even	6	1
3528.2.s.bj		4	21.h	odd	6	1
7056.2.a.ct		2	84.j	odd	6	1
7056.2.a.ct		2	84.n	even	6	1
9800.2.a.bv		2	35.i	odd	6	1
9800.2.a.bv		2	35.j	even	6	1

Hecke kernels

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

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

\( T_{5}^{4} + 8T_{5}^{2} + 64 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} + 2T^{2} + 4 \)
$5$	\( T^{4} + 8T^{2} + 64 \)
$7$	\( T^{4} \)
$11$	\( (T^{2} + 6 T + 36)^{2} \)