Newform orbit 392.4.i.b

• Label: 392.4.i.b
• Level: $392$
• Weight: $4$
• Character orbit: 392.i
• Analytic conductor: $23.129$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.1287487223\)
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.

\(f(q)\)	\(=\)	\( q + (4 \zeta_{6} - 4) q^{3} - 2 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 2 q^{5} + 11 q^{9}+O(q^{10}) \)

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

−2.00000

−

3.46410i

0

−1.00000

+

1.73205i

0

0

0

5.50000

−

9.52628i

0

361.1

0

−2.00000

+

3.46410i

0

−1.00000

−

1.73205i

0

0

0

5.50000

+

9.52628i

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	392.4.i.b		2
7.b	odd	2	1		392.4.i.g		2
7.c	even	3	1		392.4.a.e		1
7.c	even	3	1	inner	392.4.i.b		2
7.d	odd	6	1		8.4.a.a	✓	1
7.d	odd	6	1		392.4.i.g		2
21.g	even	6	1		72.4.a.c		1
28.f	even	6	1		16.4.a.a		1
28.g	odd	6	1		784.4.a.e		1
35.i	odd	6	1		200.4.a.g		1
35.k	even	12	2		200.4.c.e		2
56.j	odd	6	1		64.4.a.d		1
56.m	even	6	1		64.4.a.b		1
63.i	even	6	1		648.4.i.e		2
63.k	odd	6	1		648.4.i.h		2
63.s	even	6	1		648.4.i.e		2
63.t	odd	6	1		648.4.i.h		2
77.i	even	6	1		968.4.a.a		1
84.j	odd	6	1		144.4.a.e		1
91.s	odd	6	1		1352.4.a.a		1
105.p	even	6	1		1800.4.a.d		1
105.w	odd	12	2		1800.4.f.u		2
112.v	even	12	2		256.4.b.g		2
112.x	odd	12	2		256.4.b.a		2
119.h	odd	6	1		2312.4.a.a		1
140.s	even	6	1		400.4.a.g		1
140.x	odd	12	2		400.4.c.i		2
168.ba	even	6	1		576.4.a.k		1
168.be	odd	6	1		576.4.a.j		1
280.ba	even	6	1		1600.4.a.bm		1
280.bk	odd	6	1		1600.4.a.o		1
308.m	odd	6	1		1936.4.a.l		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.4.a.a	✓	1	7.d	odd	6	1
16.4.a.a		1	28.f	even	6	1
64.4.a.b		1	56.m	even	6	1
64.4.a.d		1	56.j	odd	6	1
72.4.a.c		1	21.g	even	6	1
144.4.a.e		1	84.j	odd	6	1
200.4.a.g		1	35.i	odd	6	1
200.4.c.e		2	35.k	even	12	2
256.4.b.a		2	112.x	odd	12	2
256.4.b.g		2	112.v	even	12	2
392.4.a.e		1	7.c	even	3	1
392.4.i.b		2	1.a	even	1	1	trivial
392.4.i.b		2	7.c	even	3	1	inner
392.4.i.g		2	7.b	odd	2	1
392.4.i.g		2	7.d	odd	6	1
400.4.a.g		1	140.s	even	6	1
400.4.c.i		2	140.x	odd	12	2
576.4.a.j		1	168.be	odd	6	1
576.4.a.k		1	168.ba	even	6	1
648.4.i.e		2	63.i	even	6	1
648.4.i.e		2	63.s	even	6	1
648.4.i.h		2	63.k	odd	6	1
648.4.i.h		2	63.t	odd	6	1
784.4.a.e		1	28.g	odd	6	1
968.4.a.a		1	77.i	even	6	1
1352.4.a.a		1	91.s	odd	6	1
1600.4.a.o		1	280.bk	odd	6	1
1600.4.a.bm		1	280.ba	even	6	1
1800.4.a.d		1	105.p	even	6	1
1800.4.f.u		2	105.w	odd	12	2
1936.4.a.l		1	308.m	odd	6	1
2312.4.a.a		1	119.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 16 \)

\( T_{5}^{2} + 2T_{5} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 4T + 16 \)
$5$	\( T^{2} + 2T + 4 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 44T + 1936 \)