Newform orbit 1008.1.f.a

• Label: 1008.1.f.a
• Level: $1008$
• Weight: $1$
• Character orbit: 1008.f
• Self dual: yes
• Analytic conductor: $0.503$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -3, -7, 21
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1008.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.503057532734\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 63)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.3024.2
Stark unit:	Root of $x^{4} - 292x^{3} - 186x^{2} - 292x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{7} + q^{25} - 2 q^{37} + 2 q^{43} + q^{49} - 2 q^{67} - 2 q^{79}+O(q^{100}) \)

Character values

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

\(n\)	\(127\)	\(577\)	\(757\)	\(785\)
\(\chi(n)\)	\(0\)	\(1\)	\(0\)	\(0\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

433.1

0

0

0

0

0

0

1.00000

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.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
21.c	even	2	1	RM by \(\Q(\sqrt{21}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.1.f.a		1
3.b	odd	2	1	CM	1008.1.f.a		1
4.b	odd	2	1		63.1.d.a	✓	1
7.b	odd	2	1	CM	1008.1.f.a		1
12.b	even	2	1		63.1.d.a	✓	1
20.d	odd	2	1		1575.1.h.b		1
20.e	even	4	2		1575.1.e.b		2
21.c	even	2	1	RM	1008.1.f.a		1
28.d	even	2	1		63.1.d.a	✓	1
28.f	even	6	2		441.1.m.a		2
28.g	odd	6	2		441.1.m.a		2
36.f	odd	6	2		567.1.l.b		2
36.h	even	6	2		567.1.l.b		2
60.h	even	2	1		1575.1.h.b		1
60.l	odd	4	2		1575.1.e.b		2
84.h	odd	2	1		63.1.d.a	✓	1
84.j	odd	6	2		441.1.m.a		2
84.n	even	6	2		441.1.m.a		2
140.c	even	2	1		1575.1.h.b		1
140.j	odd	4	2		1575.1.e.b		2
252.n	even	6	2		3969.1.k.b		2
252.o	even	6	2		3969.1.k.b		2
252.r	odd	6	2		3969.1.t.c		2
252.s	odd	6	2		567.1.l.b		2
252.u	odd	6	2		3969.1.t.c		2
252.bb	even	6	2		3969.1.t.c		2
252.bi	even	6	2		567.1.l.b		2
252.bj	even	6	2		3969.1.t.c		2
252.bl	odd	6	2		3969.1.k.b		2
252.bn	odd	6	2		3969.1.k.b		2
420.o	odd	2	1		1575.1.h.b		1
420.w	even	4	2		1575.1.e.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.1.d.a	✓	1	4.b	odd	2	1
63.1.d.a	✓	1	12.b	even	2	1
63.1.d.a	✓	1	28.d	even	2	1
63.1.d.a	✓	1	84.h	odd	2	1
441.1.m.a		2	28.f	even	6	2
441.1.m.a		2	28.g	odd	6	2
441.1.m.a		2	84.j	odd	6	2
441.1.m.a		2	84.n	even	6	2
567.1.l.b		2	36.f	odd	6	2
567.1.l.b		2	36.h	even	6	2
567.1.l.b		2	252.s	odd	6	2
567.1.l.b		2	252.bi	even	6	2
1008.1.f.a		1	1.a	even	1	1	trivial
1008.1.f.a		1	3.b	odd	2	1	CM
1008.1.f.a		1	7.b	odd	2	1	CM
1008.1.f.a		1	21.c	even	2	1	RM
1575.1.e.b		2	20.e	even	4	2
1575.1.e.b		2	60.l	odd	4	2
1575.1.e.b		2	140.j	odd	4	2
1575.1.e.b		2	420.w	even	4	2
1575.1.h.b		1	20.d	odd	2	1
1575.1.h.b		1	60.h	even	2	1
1575.1.h.b		1	140.c	even	2	1
1575.1.h.b		1	420.o	odd	2	1
3969.1.k.b		2	252.n	even	6	2
3969.1.k.b		2	252.o	even	6	2
3969.1.k.b		2	252.bl	odd	6	2
3969.1.k.b		2	252.bn	odd	6	2
3969.1.t.c		2	252.r	odd	6	2
3969.1.t.c		2	252.u	odd	6	2
3969.1.t.c		2	252.bb	even	6	2
3969.1.t.c		2	252.bj	even	6	2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T - 1 \)
$11$	\( T \)