Newform orbit 392.3.k.a

• Label: 392.3.k.a
• Level: $392$
• Weight: $3$
• Character orbit: 392.k
• Analytic conductor: $10.681$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(10.6812263629\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 2 q^{2} + \zeta_{6} q^{3} + 4 q^{4} + (3 \zeta_{6} + 3) q^{5} - 2 \zeta_{6} q^{6} - 8 q^{8} + ( - 8 \zeta_{6} + 8) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + q^{3} + 8 q^{4} + 9 q^{5} - 2 q^{6} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−2.00000

0.500000

−

0.866025i

4.00000

4.50000

−

2.59808i

−1.00000

+

1.73205i

0

−8.00000

4.00000

+

6.92820i

−9.00000

+

5.19615i

275.1

−2.00000

0.500000

+

0.866025i

4.00000

4.50000

+

2.59808i

−1.00000

−

1.73205i

0

−8.00000

4.00000

−

6.92820i

−9.00000

−

5.19615i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
56.k	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	392.3.k.a		2
7.b	odd	2	1		56.3.k.a	✓	2
7.c	even	3	1		392.3.g.d		2
7.c	even	3	1		392.3.k.c		2
7.d	odd	6	1		56.3.k.b	yes	2
7.d	odd	6	1		392.3.g.e		2
8.d	odd	2	1		392.3.k.c		2
28.d	even	2	1		224.3.o.a		2
28.f	even	6	1		224.3.o.b		2
28.f	even	6	1		1568.3.g.c		2
28.g	odd	6	1		1568.3.g.f		2
56.e	even	2	1		56.3.k.b	yes	2
56.h	odd	2	1		224.3.o.b		2
56.j	odd	6	1		224.3.o.a		2
56.j	odd	6	1		1568.3.g.c		2
56.k	odd	6	1		392.3.g.d		2
56.k	odd	6	1	inner	392.3.k.a		2
56.m	even	6	1		56.3.k.a	✓	2
56.m	even	6	1		392.3.g.e		2
56.p	even	6	1		1568.3.g.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.3.k.a	✓	2	7.b	odd	2	1
56.3.k.a	✓	2	56.m	even	6	1
56.3.k.b	yes	2	7.d	odd	6	1
56.3.k.b	yes	2	56.e	even	2	1
224.3.o.a		2	28.d	even	2	1
224.3.o.a		2	56.j	odd	6	1
224.3.o.b		2	28.f	even	6	1
224.3.o.b		2	56.h	odd	2	1
392.3.g.d		2	7.c	even	3	1
392.3.g.d		2	56.k	odd	6	1
392.3.g.e		2	7.d	odd	6	1
392.3.g.e		2	56.m	even	6	1
392.3.k.a		2	1.a	even	1	1	trivial
392.3.k.a		2	56.k	odd	6	1	inner
392.3.k.c		2	7.c	even	3	1
392.3.k.c		2	8.d	odd	2	1
1568.3.g.c		2	28.f	even	6	1
1568.3.g.c		2	56.j	odd	6	1
1568.3.g.f		2	28.g	odd	6	1
1568.3.g.f		2	56.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)

\( T_{5}^{2} - 9T_{5} + 27 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 2)^{2} \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} - 9T + 27 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 17T + 289 \)