Newform orbit 392.6.i.e

• Label: 392.6.i.e
• Level: $392$
• Weight: $6$
• Character orbit: 392.i
• Analytic conductor: $62.870$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(62.8704573667\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - 20 \zeta_{6} + 20) q^{3} - 74 \zeta_{6} q^{5} - 157 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{3} - 74 q^{5} - 157 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

177.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

10.0000

+

17.3205i

0

−37.0000

+

64.0859i

0

0

0

−78.5000

+

135.966i

0

361.1

0

10.0000

−

17.3205i

0

−37.0000

−

64.0859i

0

0

0

−78.5000

−

135.966i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.6.i.e		2
7.b	odd	2	1		392.6.i.b		2
7.c	even	3	1		392.6.a.b		1
7.c	even	3	1	inner	392.6.i.e		2
7.d	odd	6	1		8.6.a.a	✓	1
7.d	odd	6	1		392.6.i.b		2
21.g	even	6	1		72.6.a.f		1
28.f	even	6	1		16.6.a.a		1
28.g	odd	6	1		784.6.a.l		1
35.i	odd	6	1		200.6.a.a		1
35.k	even	12	2		200.6.c.a		2
56.j	odd	6	1		64.6.a.a		1
56.m	even	6	1		64.6.a.g		1
77.i	even	6	1		968.6.a.a		1
84.j	odd	6	1		144.6.a.k		1
112.v	even	12	2		256.6.b.d		2
112.x	odd	12	2		256.6.b.f		2
140.s	even	6	1		400.6.a.l		1
140.x	odd	12	2		400.6.c.d		2
168.ba	even	6	1		576.6.a.g		1
168.be	odd	6	1		576.6.a.h		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.6.a.a	✓	1	7.d	odd	6	1
16.6.a.a		1	28.f	even	6	1
64.6.a.a		1	56.j	odd	6	1
64.6.a.g		1	56.m	even	6	1
72.6.a.f		1	21.g	even	6	1
144.6.a.k		1	84.j	odd	6	1
200.6.a.a		1	35.i	odd	6	1
200.6.c.a		2	35.k	even	12	2
256.6.b.d		2	112.v	even	12	2
256.6.b.f		2	112.x	odd	12	2
392.6.a.b		1	7.c	even	3	1
392.6.i.b		2	7.b	odd	2	1
392.6.i.b		2	7.d	odd	6	1
392.6.i.e		2	1.a	even	1	1	trivial
392.6.i.e		2	7.c	even	3	1	inner
400.6.a.l		1	140.s	even	6	1
400.6.c.d		2	140.x	odd	12	2
576.6.a.g		1	168.ba	even	6	1
576.6.a.h		1	168.be	odd	6	1
784.6.a.l		1	28.g	odd	6	1
968.6.a.a		1	77.i	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20T_{3} + 400 \) acting on \(S_{6}^{\mathrm{new}}(392, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 20T + 400 \)
$5$	\( T^{2} + 74T + 5476 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 124T + 15376 \)