Newform orbit 392.2.m.c

• Label: 392.2.m.c
• Level: $392$
• Weight: $2$
• Character orbit: 392.m
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.339738624.1
Defining polynomial:	\( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{2} + ( - \beta_{7} + \beta_{5} - \beta_{3}) q^{3} + ( - 2 \beta_{4} - 2) q^{4} + (2 \beta_{5} - \beta_1) q^{5} + 2 \beta_1 q^{6} + 2 \beta_{2} q^{8} + ( - 2 \beta_{6} + \beta_{4} - 2 \beta_{2} + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + 20 ) / 14 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} + 34\nu ) / 14 \)
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \)
\(\beta_{5}\)	\(=\)	\( ( -2\nu^{7} + 7\nu^{5} - 28\nu^{3} + 16\nu ) / 14 \)
\(\beta_{6}\)	\(=\)	\( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \)
\(\beta_{7}\)	\(=\)	\( ( -6\nu^{7} + 21\nu^{5} - 70\nu^{3} + 6\nu ) / 14 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1 + \beta_{4}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

0.662827	+	0.382683i
−0.662827	−	0.382683i
−1.60021	−	0.923880i
1.60021	+	0.923880i
0.662827	−	0.382683i
−0.662827	+	0.382683i
−1.60021	+	0.923880i
1.60021	−	0.923880i

−0.707107

−

1.22474i

−0.937379

+

0.541196i

−1.00000

+

1.73205i

0.662827

−

1.14805i

1.32565

+

0.765367i

0

2.82843

−0.914214

+

1.58346i

−1.87476

19.2

−0.707107

−

1.22474i

0.937379

−

0.541196i

−1.00000

+

1.73205i

−0.662827

+

1.14805i

−1.32565

−

0.765367i

0

2.82843

−0.914214

+

1.58346i

1.87476

19.3

0.707107

+

1.22474i

−2.26303

+

1.30656i

−1.00000

+

1.73205i

−1.60021

+

2.77164i

−3.20041

−

1.84776i

0

−2.82843

1.91421

−

3.31552i

−4.52607

19.4

0.707107

+

1.22474i

2.26303

−

1.30656i

−1.00000

+

1.73205i

1.60021

−

2.77164i

3.20041

+

1.84776i

0

−2.82843

1.91421

−

3.31552i

4.52607

227.1

−0.707107

+

1.22474i

−0.937379

−

0.541196i

−1.00000

−

1.73205i

0.662827

+

1.14805i

1.32565

−

0.765367i

0

2.82843

−0.914214

−

1.58346i

−1.87476

227.2

−0.707107

+

1.22474i

0.937379

+

0.541196i

−1.00000

−

1.73205i

−0.662827

−

1.14805i

−1.32565

+

0.765367i

0

2.82843

−0.914214

−

1.58346i

1.87476

227.3

0.707107

−

1.22474i

−2.26303

−

1.30656i

−1.00000

−

1.73205i

−1.60021

−

2.77164i

−3.20041

+

1.84776i

0

−2.82843

1.91421

+

3.31552i

−4.52607

227.4

0.707107

−

1.22474i

2.26303

+

1.30656i

−1.00000

−

1.73205i

1.60021

+

2.77164i

3.20041

−

1.84776i

0

−2.82843

1.91421

+

3.31552i

4.52607

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
56.k	odd	6	1	inner
56.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.2.m.c		8
4.b	odd	2	1		1568.2.q.c		8
7.b	odd	2	1	inner	392.2.m.c		8
7.c	even	3	1		392.2.e.c	✓	8
7.c	even	3	1		392.2.m.e		8
7.d	odd	6	1		392.2.e.c	✓	8
7.d	odd	6	1		392.2.m.e		8
8.b	even	2	1		1568.2.q.d		8
8.d	odd	2	1		392.2.m.e		8
28.d	even	2	1		1568.2.q.c		8
28.f	even	6	1		1568.2.e.c		8
28.f	even	6	1		1568.2.q.d		8
28.g	odd	6	1		1568.2.e.c		8
28.g	odd	6	1		1568.2.q.d		8
56.e	even	2	1		392.2.m.e		8
56.h	odd	2	1		1568.2.q.d		8
56.j	odd	6	1		1568.2.e.c		8
56.j	odd	6	1		1568.2.q.c		8
56.k	odd	6	1		392.2.e.c	✓	8
56.k	odd	6	1	inner	392.2.m.c		8
56.m	even	6	1		392.2.e.c	✓	8
56.m	even	6	1	inner	392.2.m.c		8
56.p	even	6	1		1568.2.e.c		8
56.p	even	6	1		1568.2.q.c		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
392.2.e.c	✓	8	7.c	even	3	1
392.2.e.c	✓	8	7.d	odd	6	1
392.2.e.c	✓	8	56.k	odd	6	1
392.2.e.c	✓	8	56.m	even	6	1
392.2.m.c		8	1.a	even	1	1	trivial
392.2.m.c		8	7.b	odd	2	1	inner
392.2.m.c		8	56.k	odd	6	1	inner
392.2.m.c		8	56.m	even	6	1	inner
392.2.m.e		8	7.c	even	3	1
392.2.m.e		8	7.d	odd	6	1
392.2.m.e		8	8.d	odd	2	1
392.2.m.e		8	56.e	even	2	1
1568.2.e.c		8	28.f	even	6	1
1568.2.e.c		8	28.g	odd	6	1
1568.2.e.c		8	56.j	odd	6	1
1568.2.e.c		8	56.p	even	6	1
1568.2.q.c		8	4.b	odd	2	1
1568.2.q.c		8	28.d	even	2	1
1568.2.q.c		8	56.j	odd	6	1
1568.2.q.c		8	56.p	even	6	1
1568.2.q.d		8	8.b	even	2	1
1568.2.q.d		8	28.f	even	6	1
1568.2.q.d		8	28.g	odd	6	1
1568.2.q.d		8	56.h	odd	2	1

Hecke kernels

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

\( T_{3}^{8} - 8T_{3}^{6} + 56T_{3}^{4} - 64T_{3}^{2} + 64 \)

\( T_{5}^{8} + 12T_{5}^{6} + 126T_{5}^{4} + 216T_{5}^{2} + 324 \)

\( T_{23}^{4} - 12T_{23}^{3} + 36T_{23}^{2} + 144T_{23} + 144 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} + 2 T^{2} + 4)^{2} \)
$3$	T^8 - 8*T^6 + 56*T^4 - 64*T^2 + 64
$5$	T^8 + 12*T^6 + 126*T^4 + 216*T^2 + 324
$7$	\( T^{8} \)
$11$	\( (T^{2} - 2 T + 4)^{4} \)