Newform orbit 392.2.a.e

• Label: 392.2.a.e
• Level: $392$
• Weight: $2$
• Character orbit: 392.a
• Self dual: yes
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{3} + q^{5} - 2 q^{9}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

0

0

1.00000

0

1.00000

0

0

0

−2.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(7\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.2.a.e		1
3.b	odd	2	1		3528.2.a.j		1
4.b	odd	2	1		784.2.a.c		1
5.b	even	2	1		9800.2.a.s		1
7.b	odd	2	1		392.2.a.c		1
7.c	even	3	2		392.2.i.b		2
7.d	odd	6	2		56.2.i.b	✓	2
8.b	even	2	1		3136.2.a.i		1
8.d	odd	2	1		3136.2.a.t		1
12.b	even	2	1		7056.2.a.u		1
21.c	even	2	1		3528.2.a.p		1
21.g	even	6	2		504.2.s.c		2
21.h	odd	6	2		3528.2.s.q		2
28.d	even	2	1		784.2.a.h		1
28.f	even	6	2		112.2.i.a		2
28.g	odd	6	2		784.2.i.h		2
35.c	odd	2	1		9800.2.a.be		1
35.i	odd	6	2		1400.2.q.d		2
35.k	even	12	4		1400.2.bh.a		4
56.e	even	2	1		3136.2.a.j		1
56.h	odd	2	1		3136.2.a.u		1
56.j	odd	6	2		448.2.i.b		2
56.m	even	6	2		448.2.i.d		2
84.h	odd	2	1		7056.2.a.bj		1
84.j	odd	6	2		1008.2.s.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.i.b	✓	2	7.d	odd	6	2
112.2.i.a		2	28.f	even	6	2
392.2.a.c		1	7.b	odd	2	1
392.2.a.e		1	1.a	even	1	1	trivial
392.2.i.b		2	7.c	even	3	2
448.2.i.b		2	56.j	odd	6	2
448.2.i.d		2	56.m	even	6	2
504.2.s.c		2	21.g	even	6	2
784.2.a.c		1	4.b	odd	2	1
784.2.a.h		1	28.d	even	2	1
784.2.i.h		2	28.g	odd	6	2
1008.2.s.g		2	84.j	odd	6	2
1400.2.q.d		2	35.i	odd	6	2
1400.2.bh.a		4	35.k	even	12	4
3136.2.a.i		1	8.b	even	2	1
3136.2.a.j		1	56.e	even	2	1
3136.2.a.t		1	8.d	odd	2	1
3136.2.a.u		1	56.h	odd	2	1
3528.2.a.j		1	3.b	odd	2	1
3528.2.a.p		1	21.c	even	2	1
3528.2.s.q		2	21.h	odd	6	2
7056.2.a.u		1	12.b	even	2	1
7056.2.a.bj		1	84.h	odd	2	1
9800.2.a.s		1	5.b	even	2	1
9800.2.a.be		1	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(392))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T - 1 \)
$5$	\( T - 1 \)
$7$	\( T \)
$11$	\( T - 3 \)