Newform orbit 392.3.j.c

• Label: 392.3.j.c
• Level: $392$
• Weight: $3$
• Character orbit: 392.j
• Analytic conductor: $10.681$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -56
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(10.6812263629\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial:	\( x^{4} + 7x^{2} + 49 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 56)
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.

\(f(q)\)	\(=\)	\( q - 2 \beta_{2} q^{2} + \beta_1 q^{3} + ( - 4 \beta_{2} - 4) q^{4} + (\beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} - 8 q^{8} + 19 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 38 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) :

\(\beta_{1}\)	\(=\)	\( 2\nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 7 \)
\(\beta_{3}\)	\(=\)	\( ( 2\nu^{3} ) / 7 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

117.1

−1.32288	−	2.29129i
1.32288	+	2.29129i
−1.32288	+	2.29129i
1.32288	−	2.29129i

1.00000

−

1.73205i

−2.64575

−

4.58258i

−2.00000

−

3.46410i

2.64575

−

4.58258i

−10.5830

0

−8.00000

−9.50000

+

16.4545i

−5.29150

−

9.16515i

117.2

1.00000

−

1.73205i

2.64575

+

4.58258i

−2.00000

−

3.46410i

−2.64575

+

4.58258i

10.5830

0

−8.00000

−9.50000

+

16.4545i

5.29150

+

9.16515i

325.1

1.00000

+

1.73205i

−2.64575

+

4.58258i

−2.00000

+

3.46410i

2.64575

+

4.58258i

−10.5830

0

−8.00000

−9.50000

−

16.4545i

−5.29150

+

9.16515i

325.2

1.00000

+

1.73205i

2.64575

−

4.58258i

−2.00000

+

3.46410i

−2.64575

−

4.58258i

10.5830

0

−8.00000

−9.50000

−

16.4545i

5.29150

−

9.16515i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.h	odd	2	1	CM by \(\Q(\sqrt{-14}) \)
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
8.b	even	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	392.3.j.c		4
7.b	odd	2	1	inner	392.3.j.c		4
7.c	even	3	1		56.3.h.a	✓	2
7.c	even	3	1	inner	392.3.j.c		4
7.d	odd	6	1		56.3.h.a	✓	2
7.d	odd	6	1	inner	392.3.j.c		4
8.b	even	2	1	inner	392.3.j.c		4
21.g	even	6	1		504.3.l.c		2
21.h	odd	6	1		504.3.l.c		2
28.f	even	6	1		224.3.h.b		2
28.g	odd	6	1		224.3.h.b		2
56.h	odd	2	1	CM	392.3.j.c		4
56.j	odd	6	1		56.3.h.a	✓	2
56.j	odd	6	1	inner	392.3.j.c		4
56.k	odd	6	1		224.3.h.b		2
56.m	even	6	1		224.3.h.b		2
56.p	even	6	1		56.3.h.a	✓	2
56.p	even	6	1	inner	392.3.j.c		4
84.j	odd	6	1		2016.3.l.a		2
84.n	even	6	1		2016.3.l.a		2
168.s	odd	6	1		504.3.l.c		2
168.v	even	6	1		2016.3.l.a		2
168.ba	even	6	1		504.3.l.c		2
168.be	odd	6	1		2016.3.l.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.3.h.a	✓	2	7.c	even	3	1
56.3.h.a	✓	2	7.d	odd	6	1
56.3.h.a	✓	2	56.j	odd	6	1
56.3.h.a	✓	2	56.p	even	6	1
224.3.h.b		2	28.f	even	6	1
224.3.h.b		2	28.g	odd	6	1
224.3.h.b		2	56.k	odd	6	1
224.3.h.b		2	56.m	even	6	1
392.3.j.c		4	1.a	even	1	1	trivial
392.3.j.c		4	7.b	odd	2	1	inner
392.3.j.c		4	7.c	even	3	1	inner
392.3.j.c		4	7.d	odd	6	1	inner
392.3.j.c		4	8.b	even	2	1	inner
392.3.j.c		4	56.h	odd	2	1	CM
392.3.j.c		4	56.j	odd	6	1	inner
392.3.j.c		4	56.p	even	6	1	inner
504.3.l.c		2	21.g	even	6	1
504.3.l.c		2	21.h	odd	6	1
504.3.l.c		2	168.s	odd	6	1
504.3.l.c		2	168.ba	even	6	1
2016.3.l.a		2	84.j	odd	6	1
2016.3.l.a		2	84.n	even	6	1
2016.3.l.a		2	168.v	even	6	1
2016.3.l.a		2	168.be	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 28T_{3}^{2} + 784 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\).

Hecke characteristic polynomials

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