Newform orbit 392.3.g.a

• Label: 392.3.g.a
• Level: $392$
• Weight: $3$
• Character orbit: 392.g
• Self dual: yes
• Analytic conductor: $10.681$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.6812263629\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 2 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{8} - 5 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

99.1

0

−2.00000

2.00000

4.00000

0

−4.00000

0

−8.00000

−5.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.3.g.a		1
4.b	odd	2	1		1568.3.g.a		1
7.b	odd	2	1		8.3.d.a	✓	1
7.c	even	3	2		392.3.k.b		2
7.d	odd	6	2		392.3.k.d		2
8.b	even	2	1		1568.3.g.a		1
8.d	odd	2	1	CM	392.3.g.a		1
21.c	even	2	1		72.3.b.a		1
28.d	even	2	1		32.3.d.a		1
35.c	odd	2	1		200.3.g.a		1
35.f	even	4	2		200.3.e.a		2
56.e	even	2	1		8.3.d.a	✓	1
56.h	odd	2	1		32.3.d.a		1
56.k	odd	6	2		392.3.k.b		2
56.m	even	6	2		392.3.k.d		2
84.h	odd	2	1		288.3.b.a		1
112.j	even	4	2		256.3.c.e		2
112.l	odd	4	2		256.3.c.e		2
140.c	even	2	1		800.3.g.a		1
140.j	odd	4	2		800.3.e.a		2
168.e	odd	2	1		72.3.b.a		1
168.i	even	2	1		288.3.b.a		1
280.c	odd	2	1		800.3.g.a		1
280.n	even	2	1		200.3.g.a		1
280.s	even	4	2		800.3.e.a		2
280.y	odd	4	2		200.3.e.a		2
336.v	odd	4	2		2304.3.g.j		2
336.y	even	4	2		2304.3.g.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.3.d.a	✓	1	7.b	odd	2	1
8.3.d.a	✓	1	56.e	even	2	1
32.3.d.a		1	28.d	even	2	1
32.3.d.a		1	56.h	odd	2	1
72.3.b.a		1	21.c	even	2	1
72.3.b.a		1	168.e	odd	2	1
200.3.e.a		2	35.f	even	4	2
200.3.e.a		2	280.y	odd	4	2
200.3.g.a		1	35.c	odd	2	1
200.3.g.a		1	280.n	even	2	1
256.3.c.e		2	112.j	even	4	2
256.3.c.e		2	112.l	odd	4	2
288.3.b.a		1	84.h	odd	2	1
288.3.b.a		1	168.i	even	2	1
392.3.g.a		1	1.a	even	1	1	trivial
392.3.g.a		1	8.d	odd	2	1	CM
392.3.k.b		2	7.c	even	3	2
392.3.k.b		2	56.k	odd	6	2
392.3.k.d		2	7.d	odd	6	2
392.3.k.d		2	56.m	even	6	2
800.3.e.a		2	140.j	odd	4	2
800.3.e.a		2	280.s	even	4	2
800.3.g.a		1	140.c	even	2	1
800.3.g.a		1	280.c	odd	2	1
1568.3.g.a		1	4.b	odd	2	1
1568.3.g.a		1	8.b	even	2	1
2304.3.g.j		2	336.v	odd	4	2
2304.3.g.j		2	336.y	even	4	2

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 2 \)
$3$	\( T - 2 \)
$5$	\( T \)
$7$	\( T \)
$11$	\( T - 14 \)