Newform orbit 1008.2.s.j

• Label: 1008.2.s.j
• Level: $1008$
• Weight: $2$
• Character orbit: 1008.s
• Analytic conductor: $8.049$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.04892052375\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + ( - 3 \zeta_{6} + 2) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(577\)	\(757\)	\(785\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

289.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

0

0

0.500000

+

2.59808i

0

0

0

865.1

0

0

0

0

0

0.500000

−

2.59808i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.2.s.j		2
3.b	odd	2	1	CM	1008.2.s.j		2
4.b	odd	2	1		63.2.e.a	✓	2
7.c	even	3	1	inner	1008.2.s.j		2
7.c	even	3	1		7056.2.a.y		1
7.d	odd	6	1		7056.2.a.bf		1
12.b	even	2	1		63.2.e.a	✓	2
21.g	even	6	1		7056.2.a.bf		1
21.h	odd	6	1	inner	1008.2.s.j		2
21.h	odd	6	1		7056.2.a.y		1
28.d	even	2	1		441.2.e.c		2
28.f	even	6	1		441.2.a.e		1
28.f	even	6	1		441.2.e.c		2
28.g	odd	6	1		63.2.e.a	✓	2
28.g	odd	6	1		441.2.a.d		1
36.f	odd	6	1		567.2.g.d		2
36.f	odd	6	1		567.2.h.c		2
36.h	even	6	1		567.2.g.d		2
36.h	even	6	1		567.2.h.c		2
84.h	odd	2	1		441.2.e.c		2
84.j	odd	6	1		441.2.a.e		1
84.j	odd	6	1		441.2.e.c		2
84.n	even	6	1		63.2.e.a	✓	2
84.n	even	6	1		441.2.a.d		1
252.o	even	6	1		567.2.h.c		2
252.u	odd	6	1		567.2.g.d		2
252.bb	even	6	1		567.2.g.d		2
252.bl	odd	6	1		567.2.h.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.e.a	✓	2	4.b	odd	2	1
63.2.e.a	✓	2	12.b	even	2	1
63.2.e.a	✓	2	28.g	odd	6	1
63.2.e.a	✓	2	84.n	even	6	1
441.2.a.d		1	28.g	odd	6	1
441.2.a.d		1	84.n	even	6	1
441.2.a.e		1	28.f	even	6	1
441.2.a.e		1	84.j	odd	6	1
441.2.e.c		2	28.d	even	2	1
441.2.e.c		2	28.f	even	6	1
441.2.e.c		2	84.h	odd	2	1
441.2.e.c		2	84.j	odd	6	1
567.2.g.d		2	36.f	odd	6	1
567.2.g.d		2	36.h	even	6	1
567.2.g.d		2	252.u	odd	6	1
567.2.g.d		2	252.bb	even	6	1
567.2.h.c		2	36.f	odd	6	1
567.2.h.c		2	36.h	even	6	1
567.2.h.c		2	252.o	even	6	1
567.2.h.c		2	252.bl	odd	6	1
1008.2.s.j		2	1.a	even	1	1	trivial
1008.2.s.j		2	3.b	odd	2	1	CM
1008.2.s.j		2	7.c	even	3	1	inner
1008.2.s.j		2	21.h	odd	6	1	inner
7056.2.a.y		1	7.c	even	3	1
7056.2.a.y		1	21.h	odd	6	1
7056.2.a.bf		1	7.d	odd	6	1
7056.2.a.bf		1	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5} \)

\( T_{11} \)

\( T_{13} + 7 \)

\( T_{17} \)

Hecke characteristic polynomials

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