Newform orbit 392.1.g.b

• Label: 392.1.g.b
• Level: $392$
• Weight: $1$
• Character orbit: 392.g
• Self dual: yes
• Analytic conductor: $0.196$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.195633484952\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.0.2744.1
Artin image:	$D_8$
Artin field:	Galois closure of 8.0.421654016.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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

1.41421
−1.41421

−1.00000

−1.41421

1.00000

0

1.41421

0

−1.00000

1.00000

0

99.2

−1.00000

1.41421

1.00000

0

−1.41421

0

−1.00000

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
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	392.1.g.b	✓	2
3.b	odd	2	1		3528.1.g.c		2
4.b	odd	2	1		1568.1.g.b		2
7.b	odd	2	1	inner	392.1.g.b	✓	2
7.c	even	3	2		392.1.k.b		4
7.d	odd	6	2		392.1.k.b		4
8.b	even	2	1		1568.1.g.b		2
8.d	odd	2	1	CM	392.1.g.b	✓	2
21.c	even	2	1		3528.1.g.c		2
21.g	even	6	2		3528.1.bx.b		4
21.h	odd	6	2		3528.1.bx.b		4
24.f	even	2	1		3528.1.g.c		2
28.d	even	2	1		1568.1.g.b		2
28.f	even	6	2		1568.1.o.b		4
28.g	odd	6	2		1568.1.o.b		4
56.e	even	2	1	inner	392.1.g.b	✓	2
56.h	odd	2	1		1568.1.g.b		2
56.j	odd	6	2		1568.1.o.b		4
56.k	odd	6	2		392.1.k.b		4
56.m	even	6	2		392.1.k.b		4
56.p	even	6	2		1568.1.o.b		4
168.e	odd	2	1		3528.1.g.c		2
168.v	even	6	2		3528.1.bx.b		4
168.be	odd	6	2		3528.1.bx.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.1.g.b	✓	2	1.a	even	1	1	trivial
392.1.g.b	✓	2	7.b	odd	2	1	inner
392.1.g.b	✓	2	8.d	odd	2	1	CM
392.1.g.b	✓	2	56.e	even	2	1	inner
392.1.k.b		4	7.c	even	3	2
392.1.k.b		4	7.d	odd	6	2
392.1.k.b		4	56.k	odd	6	2
392.1.k.b		4	56.m	even	6	2
1568.1.g.b		2	4.b	odd	2	1
1568.1.g.b		2	8.b	even	2	1
1568.1.g.b		2	28.d	even	2	1
1568.1.g.b		2	56.h	odd	2	1
1568.1.o.b		4	28.f	even	6	2
1568.1.o.b		4	28.g	odd	6	2
1568.1.o.b		4	56.j	odd	6	2
1568.1.o.b		4	56.p	even	6	2
3528.1.g.c		2	3.b	odd	2	1
3528.1.g.c		2	21.c	even	2	1
3528.1.g.c		2	24.f	even	2	1
3528.1.g.c		2	168.e	odd	2	1
3528.1.bx.b		4	21.g	even	6	2
3528.1.bx.b		4	21.h	odd	6	2
3528.1.bx.b		4	168.v	even	6	2
3528.1.bx.b		4	168.be	odd	6	2

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} - 2 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} \)