Newform orbit 392.2.p.f

• Label: 392.2.p.f
• Level: $392$
• Weight: $2$
• Character orbit: 392.p
• 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.p (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:	8.0.432972864.2
Defining polynomial:	\( x^{8} - x^{7} + x^{6} + 4x^{5} - 6x^{4} + 8x^{3} + 4x^{2} - 8x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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 + (-b6 - b1) * q^2 + (-b6 - b5) * q^3 + b5 * q^4 + (b7 - b6 + b4 - b1) * q^5 + (b7 + b5 + b3 + 2) * q^6 + (-b7 - 2) * q^8 + (b7 + 2*b6 + b4 + b3 + 3*b2 + 2*b1 + 3) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - q^{4} + 12 q^{6} - 14 q^{8} + 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + x^{6} + 4x^{5} - 6x^{4} + 8x^{3} + 4x^{2} - 8x + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{6} - \nu^{5} + \nu^{4} - 2\nu^{2} + 4\nu - 4 ) / 4 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + \nu^{5} - 2\nu^{3} + 4\nu^{2} - 4\nu ) / 8 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + 5\nu^{5} - 2\nu^{3} + 8\nu^{2} - 4\nu + 8 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} + \nu^{5} - 6\nu^{4} - 2\nu^{3} + 4\nu^{2} - 4\nu ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} + \nu^{5} - 6\nu^{4} - 2\nu^{3} + 4\nu^{2} - 20\nu ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 2\nu^{4} - 6\nu^{3} + 12\nu^{2} - 4\nu ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{6} + \nu^{5} - \nu^{4} + 6\nu^{3} + 2\nu^{2} - 4\nu + 12 ) / 4 \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1 - \beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

165.1

−1.41156	+	0.0865986i
0.121053	−	1.40902i
0.630783	+	1.26575i
1.15972	−	0.809347i
−1.41156	−	0.0865986i
0.121053	+	1.40902i
0.630783	−	1.26575i
1.15972	+	0.809347i

−1.41156

−

0.0865986i

−2.61578

−

1.51022i

1.98500

+

0.244478i

−1.46890

+

0.848071i

3.56155

+

2.35829i

0

−2.78078

−

0.516994i

3.06155

+

5.30277i

2.14688

−

1.06990i

165.2

0.121053

+

1.40902i

0.810969

+

0.468213i

−1.97069

+

0.341134i

−2.88831

+

1.66757i

−0.561553

+

1.19935i

0

−0.719224

−

2.73546i

−1.06155

−

1.83866i

−2.69928

−

3.86783i

165.3

0.630783

−

1.26575i

2.61578

+

1.51022i

−1.20422

−

1.59682i

1.46890

−

0.848071i

3.56155

−

2.35829i

0

−2.78078

+

0.516994i

3.06155

+

5.30277i

−0.146883

−

2.39420i

165.4

1.15972

+

0.809347i

−0.810969

−

0.468213i

0.689916

+

1.87724i

2.88831

−

1.66757i

−0.561553

−

1.19935i

0

−0.719224

+

2.73546i

−1.06155

−

1.83866i

4.69928

+

0.403728i

373.1

−1.41156

+

0.0865986i

−2.61578

+

1.51022i

1.98500

−

0.244478i

−1.46890

−

0.848071i

3.56155

−

2.35829i

0

−2.78078

+

0.516994i

3.06155

−

5.30277i

2.14688

+

1.06990i

373.2

0.121053

−

1.40902i

0.810969

−

0.468213i

−1.97069

−

0.341134i

−2.88831

−

1.66757i

−0.561553

−

1.19935i

0

−0.719224

+

2.73546i

−1.06155

+

1.83866i

−2.69928

+

3.86783i

373.3

0.630783

+

1.26575i

2.61578

−

1.51022i

−1.20422

+

1.59682i

1.46890

+

0.848071i

3.56155

+

2.35829i

0

−2.78078

−

0.516994i

3.06155

−

5.30277i

−0.146883

+

2.39420i

373.4

1.15972

−

0.809347i

−0.810969

+

0.468213i

0.689916

−

1.87724i

2.88831

+

1.66757i

−0.561553

+

1.19935i

0

−0.719224

−

2.73546i

−1.06155

+

1.83866i

4.69928

−

0.403728i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
8.b	even	2	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	392.2.p.f		8
4.b	odd	2	1		1568.2.t.d		8
7.b	odd	2	1		392.2.p.e		8
7.c	even	3	1		56.2.b.b	✓	4
7.c	even	3	1	inner	392.2.p.f		8
7.d	odd	6	1		392.2.b.c		4
7.d	odd	6	1		392.2.p.e		8
8.b	even	2	1	inner	392.2.p.f		8
8.d	odd	2	1		1568.2.t.d		8
21.h	odd	6	1		504.2.c.d		4
28.d	even	2	1		1568.2.t.e		8
28.f	even	6	1		1568.2.b.d		4
28.f	even	6	1		1568.2.t.e		8
28.g	odd	6	1		224.2.b.b		4
28.g	odd	6	1		1568.2.t.d		8
56.e	even	2	1		1568.2.t.e		8
56.h	odd	2	1		392.2.p.e		8
56.j	odd	6	1		392.2.b.c		4
56.j	odd	6	1		392.2.p.e		8
56.k	odd	6	1		224.2.b.b		4
56.k	odd	6	1		1568.2.t.d		8
56.m	even	6	1		1568.2.b.d		4
56.m	even	6	1		1568.2.t.e		8
56.p	even	6	1		56.2.b.b	✓	4
56.p	even	6	1	inner	392.2.p.f		8
84.n	even	6	1		2016.2.c.c		4
112.u	odd	12	2		1792.2.a.v		4
112.w	even	12	2		1792.2.a.x		4
168.s	odd	6	1		504.2.c.d		4
168.v	even	6	1		2016.2.c.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.b.b	✓	4	7.c	even	3	1
56.2.b.b	✓	4	56.p	even	6	1
224.2.b.b		4	28.g	odd	6	1
224.2.b.b		4	56.k	odd	6	1
392.2.b.c		4	7.d	odd	6	1
392.2.b.c		4	56.j	odd	6	1
392.2.p.e		8	7.b	odd	2	1
392.2.p.e		8	7.d	odd	6	1
392.2.p.e		8	56.h	odd	2	1
392.2.p.e		8	56.j	odd	6	1
392.2.p.f		8	1.a	even	1	1	trivial
392.2.p.f		8	7.c	even	3	1	inner
392.2.p.f		8	8.b	even	2	1	inner
392.2.p.f		8	56.p	even	6	1	inner
504.2.c.d		4	21.h	odd	6	1
504.2.c.d		4	168.s	odd	6	1
1568.2.b.d		4	28.f	even	6	1
1568.2.b.d		4	56.m	even	6	1
1568.2.t.d		8	4.b	odd	2	1
1568.2.t.d		8	8.d	odd	2	1
1568.2.t.d		8	28.g	odd	6	1
1568.2.t.d		8	56.k	odd	6	1
1568.2.t.e		8	28.d	even	2	1
1568.2.t.e		8	28.f	even	6	1
1568.2.t.e		8	56.e	even	2	1
1568.2.t.e		8	56.m	even	6	1
1792.2.a.v		4	112.u	odd	12	2
1792.2.a.x		4	112.w	even	12	2
2016.2.c.c		4	84.n	even	6	1
2016.2.c.c		4	168.v	even	6	1

Hecke kernels

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

\( T_{3}^{8} - 10T_{3}^{6} + 92T_{3}^{4} - 80T_{3}^{2} + 64 \)

\( T_{17}^{2} + 2T_{17} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 - T^7 + T^6 + 4*T^5 - 6*T^4 + 8*T^3 + 4*T^2 - 8*T + 16
$3$	T^8 - 10*T^6 + 92*T^4 - 80*T^2 + 64
$5$	T^8 - 14*T^6 + 164*T^4 - 448*T^2 + 1024
$7$	\( T^{8} \)
$11$	T^8 - 20*T^6 + 368*T^4 - 640*T^2 + 1024