Newform orbit 1008.2.r.g

• Label: 1008.2.r.g
• Level: $1008$
• Weight: $2$
• Character orbit: 1008.r
• Analytic conductor: $8.049$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.04892052375\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.309123.1
Defining polynomial:	\( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b4 + b3 + b2 - 1) * q^3 + (b5 + b4 + b3 - b1) * q^5 + (-b3 + 1) * q^7 + (b5 + b3 - b2 - b1 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} + 3 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) :

\(\beta_{1}\)	\(=\)	\( \nu^{2} - 2\nu + 3 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} - 4\nu^{4} + 11\nu^{3} - 20\nu^{2} + 18\nu - 12 ) / 3 \)
\(\beta_{3}\)	\(=\)	\( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \)
\(\beta_{4}\)	\(=\)	\( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \)
\(\beta_{5}\)	\(=\)	\( -\nu^{5} + 2\nu^{4} - 8\nu^{3} + 9\nu^{2} - 13\nu + 6 \)

Character values

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

\(n\)	\(127\)	\(577\)	\(757\)	\(785\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-\beta_{3}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

0.500000	+	0.224437i
0.500000	+	2.05195i
0.500000	−	1.41036i
0.500000	−	0.224437i
0.500000	−	2.05195i
0.500000	+	1.41036i

0

−1.29418

+

1.15113i

0

0.555632

−

0.962383i

0

0.500000

+

0.866025i

0

0.349814

−

2.97954i

0

337.2

0

−0.796790

−

1.53790i

0

−1.02704

+

1.77889i

0

0.500000

+

0.866025i

0

−1.73025

+

2.45076i

0

337.3

0

1.09097

−

1.34528i

0

1.97141

−

3.41458i

0

0.500000

+

0.866025i

0

−0.619562

−

2.93533i

0

673.1

0

−1.29418

−

1.15113i

0

0.555632

+

0.962383i

0

0.500000

−

0.866025i

0

0.349814

+

2.97954i

0

673.2

0

−0.796790

+

1.53790i

0

−1.02704

−

1.77889i

0

0.500000

−

0.866025i

0

−1.73025

−

2.45076i

0

673.3

0

1.09097

+

1.34528i

0

1.97141

+

3.41458i

0

0.500000

−

0.866025i

0

−0.619562

+

2.93533i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.2.r.g		6
3.b	odd	2	1		3024.2.r.i		6
4.b	odd	2	1		252.2.j.b	✓	6
9.c	even	3	1	inner	1008.2.r.g		6
9.c	even	3	1		9072.2.a.bt		3
9.d	odd	6	1		3024.2.r.i		6
9.d	odd	6	1		9072.2.a.bz		3
12.b	even	2	1		756.2.j.a		6
28.d	even	2	1		1764.2.j.d		6
28.f	even	6	1		1764.2.i.e		6
28.f	even	6	1		1764.2.l.g		6
28.g	odd	6	1		1764.2.i.f		6
28.g	odd	6	1		1764.2.l.d		6
36.f	odd	6	1		252.2.j.b	✓	6
36.f	odd	6	1		2268.2.a.g		3
36.h	even	6	1		756.2.j.a		6
36.h	even	6	1		2268.2.a.j		3
84.h	odd	2	1		5292.2.j.e		6
84.j	odd	6	1		5292.2.i.g		6
84.j	odd	6	1		5292.2.l.d		6
84.n	even	6	1		5292.2.i.d		6
84.n	even	6	1		5292.2.l.g		6
252.n	even	6	1		1764.2.i.e		6
252.o	even	6	1		5292.2.i.d		6
252.r	odd	6	1		5292.2.l.d		6
252.s	odd	6	1		5292.2.j.e		6
252.u	odd	6	1		1764.2.l.d		6
252.bb	even	6	1		5292.2.l.g		6
252.bi	even	6	1		1764.2.j.d		6
252.bj	even	6	1		1764.2.l.g		6
252.bl	odd	6	1		1764.2.i.f		6
252.bn	odd	6	1		5292.2.i.g		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.j.b	✓	6	4.b	odd	2	1
252.2.j.b	✓	6	36.f	odd	6	1
756.2.j.a		6	12.b	even	2	1
756.2.j.a		6	36.h	even	6	1
1008.2.r.g		6	1.a	even	1	1	trivial
1008.2.r.g		6	9.c	even	3	1	inner
1764.2.i.e		6	28.f	even	6	1
1764.2.i.e		6	252.n	even	6	1
1764.2.i.f		6	28.g	odd	6	1
1764.2.i.f		6	252.bl	odd	6	1
1764.2.j.d		6	28.d	even	2	1
1764.2.j.d		6	252.bi	even	6	1
1764.2.l.d		6	28.g	odd	6	1
1764.2.l.d		6	252.u	odd	6	1
1764.2.l.g		6	28.f	even	6	1
1764.2.l.g		6	252.bj	even	6	1
2268.2.a.g		3	36.f	odd	6	1
2268.2.a.j		3	36.h	even	6	1
3024.2.r.i		6	3.b	odd	2	1
3024.2.r.i		6	9.d	odd	6	1
5292.2.i.d		6	84.n	even	6	1
5292.2.i.d		6	252.o	even	6	1
5292.2.i.g		6	84.j	odd	6	1
5292.2.i.g		6	252.bn	odd	6	1
5292.2.j.e		6	84.h	odd	2	1
5292.2.j.e		6	252.s	odd	6	1
5292.2.l.d		6	84.j	odd	6	1
5292.2.l.d		6	252.r	odd	6	1
5292.2.l.g		6	84.n	even	6	1
5292.2.l.g		6	252.bb	even	6	1
9072.2.a.bt		3	9.c	even	3	1
9072.2.a.bz		3	9.d	odd	6	1

Hecke kernels

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

\( T_{5}^{6} - 3T_{5}^{5} + 15T_{5}^{4} + 63T_{5}^{2} - 54T_{5} + 81 \)

\( T_{11}^{6} + 6T_{11}^{5} + 33T_{11}^{4} + 36T_{11}^{3} + 63T_{11}^{2} - 27T_{11} + 81 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{6} \)
$3$	T^6 + 2*T^5 + 4*T^4 + 3*T^3 + 12*T^2 + 18*T + 27
$5$	T^6 - 3*T^5 + 15*T^4 + 63*T^2 - 54*T + 81
$7$	\( (T^{2} - T + 1)^{3} \)
$11$	T^6 + 6*T^5 + 33*T^4 + 36*T^3 + 63*T^2 - 27*T + 81