Newform orbit 392.3.k.d

• Label: 392.3.k.d
• Level: $392$
• Weight: $3$
• Character orbit: 392.k
• Analytic conductor: $10.681$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -8
• Inner twists: $4$

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:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

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

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

1.00000

+

1.73205i

1.00000

−

1.73205i

−2.00000

+

3.46410i

0

4.00000

0

−8.00000

2.50000

+

4.33013i

0

275.1

1.00000

−

1.73205i

1.00000

+

1.73205i

−2.00000

−

3.46410i

0

4.00000

0

−8.00000

2.50000

−

4.33013i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
7.c	even	3	1	inner
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.d		2
7.b	odd	2	1		392.3.k.b		2
7.c	even	3	1		8.3.d.a	✓	1
7.c	even	3	1	inner	392.3.k.d		2
7.d	odd	6	1		392.3.g.a		1
7.d	odd	6	1		392.3.k.b		2
8.d	odd	2	1	CM	392.3.k.d		2
21.h	odd	6	1		72.3.b.a		1
28.f	even	6	1		1568.3.g.a		1
28.g	odd	6	1		32.3.d.a		1
35.j	even	6	1		200.3.g.a		1
35.l	odd	12	2		200.3.e.a		2
56.e	even	2	1		392.3.k.b		2
56.j	odd	6	1		1568.3.g.a		1
56.k	odd	6	1		8.3.d.a	✓	1
56.k	odd	6	1	inner	392.3.k.d		2
56.m	even	6	1		392.3.g.a		1
56.m	even	6	1		392.3.k.b		2
56.p	even	6	1		32.3.d.a		1
84.n	even	6	1		288.3.b.a		1
112.u	odd	12	2		256.3.c.e		2
112.w	even	12	2		256.3.c.e		2
140.p	odd	6	1		800.3.g.a		1
140.w	even	12	2		800.3.e.a		2
168.s	odd	6	1		288.3.b.a		1
168.v	even	6	1		72.3.b.a		1
280.bf	even	6	1		800.3.g.a		1
280.bi	odd	6	1		200.3.g.a		1
280.br	even	12	2		200.3.e.a		2
280.bt	odd	12	2		800.3.e.a		2
336.bt	odd	12	2		2304.3.g.j		2
336.bu	even	12	2		2304.3.g.j		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.3.d.a	✓	1	7.c	even	3	1
8.3.d.a	✓	1	56.k	odd	6	1
32.3.d.a		1	28.g	odd	6	1
32.3.d.a		1	56.p	even	6	1
72.3.b.a		1	21.h	odd	6	1
72.3.b.a		1	168.v	even	6	1
200.3.e.a		2	35.l	odd	12	2
200.3.e.a		2	280.br	even	12	2
200.3.g.a		1	35.j	even	6	1
200.3.g.a		1	280.bi	odd	6	1
256.3.c.e		2	112.u	odd	12	2
256.3.c.e		2	112.w	even	12	2
288.3.b.a		1	84.n	even	6	1
288.3.b.a		1	168.s	odd	6	1
392.3.g.a		1	7.d	odd	6	1
392.3.g.a		1	56.m	even	6	1
392.3.k.b		2	7.b	odd	2	1
392.3.k.b		2	7.d	odd	6	1
392.3.k.b		2	56.e	even	2	1
392.3.k.b		2	56.m	even	6	1
392.3.k.d		2	1.a	even	1	1	trivial
392.3.k.d		2	7.c	even	3	1	inner
392.3.k.d		2	8.d	odd	2	1	CM
392.3.k.d		2	56.k	odd	6	1	inner
800.3.e.a		2	140.w	even	12	2
800.3.e.a		2	280.bt	odd	12	2
800.3.g.a		1	140.p	odd	6	1
800.3.g.a		1	280.bf	even	6	1
1568.3.g.a		1	28.f	even	6	1
1568.3.g.a		1	56.j	odd	6	1
2304.3.g.j		2	336.bt	odd	12	2
2304.3.g.j		2	336.bu	even	12	2

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} + 4 \)

\( T_{5} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T + 4 \)
$3$	\( T^{2} - 2T + 4 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 14T + 196 \)