Newform orbit 567.2.i.b

• Label: 567.2.i.b
• Level: $567$
• Weight: $2$
• Character orbit: 567.i
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
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.

\(f(q)\)	\(=\)	\( q + 2 q^{4} + (2 \zeta_{6} + 1) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(\zeta_{6}\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

215.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

2.00000

0

0

2.00000

−

1.73205i

0

0

0

269.1

0

0

2.00000

0

0

2.00000

+

1.73205i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
63.i	even	6	1	inner
63.t	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	567.2.i.b		2
3.b	odd	2	1	CM	567.2.i.b		2
7.d	odd	6	1		567.2.s.a		2
9.c	even	3	1		21.2.g.a	✓	2
9.c	even	3	1		567.2.s.a		2
9.d	odd	6	1		21.2.g.a	✓	2
9.d	odd	6	1		567.2.s.a		2
21.g	even	6	1		567.2.s.a		2
36.f	odd	6	1		336.2.bc.c		2
36.h	even	6	1		336.2.bc.c		2
45.h	odd	6	1		525.2.t.c		2
45.j	even	6	1		525.2.t.c		2
45.k	odd	12	2		525.2.q.d		4
45.l	even	12	2		525.2.q.d		4
63.g	even	3	1		147.2.g.a		2
63.h	even	3	1		147.2.c.a		2
63.i	even	6	1		147.2.c.a		2
63.i	even	6	1	inner	567.2.i.b		2
63.j	odd	6	1		147.2.c.a		2
63.k	odd	6	1		21.2.g.a	✓	2
63.l	odd	6	1		147.2.g.a		2
63.n	odd	6	1		147.2.g.a		2
63.o	even	6	1		147.2.g.a		2
63.s	even	6	1		21.2.g.a	✓	2
63.t	odd	6	1		147.2.c.a		2
63.t	odd	6	1	inner	567.2.i.b		2
252.n	even	6	1		336.2.bc.c		2
252.r	odd	6	1		2352.2.k.c		2
252.u	odd	6	1		2352.2.k.c		2
252.bb	even	6	1		2352.2.k.c		2
252.bj	even	6	1		2352.2.k.c		2
252.bn	odd	6	1		336.2.bc.c		2
315.u	even	6	1		525.2.t.c		2
315.bn	odd	6	1		525.2.t.c		2
315.bw	odd	12	2		525.2.q.d		4
315.cg	even	12	2		525.2.q.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.2.g.a	✓	2	9.c	even	3	1
21.2.g.a	✓	2	9.d	odd	6	1
21.2.g.a	✓	2	63.k	odd	6	1
21.2.g.a	✓	2	63.s	even	6	1
147.2.c.a		2	63.h	even	3	1
147.2.c.a		2	63.i	even	6	1
147.2.c.a		2	63.j	odd	6	1
147.2.c.a		2	63.t	odd	6	1
147.2.g.a		2	63.g	even	3	1
147.2.g.a		2	63.l	odd	6	1
147.2.g.a		2	63.n	odd	6	1
147.2.g.a		2	63.o	even	6	1
336.2.bc.c		2	36.f	odd	6	1
336.2.bc.c		2	36.h	even	6	1
336.2.bc.c		2	252.n	even	6	1
336.2.bc.c		2	252.bn	odd	6	1
525.2.q.d		4	45.k	odd	12	2
525.2.q.d		4	45.l	even	12	2
525.2.q.d		4	315.bw	odd	12	2
525.2.q.d		4	315.cg	even	12	2
525.2.t.c		2	45.h	odd	6	1
525.2.t.c		2	45.j	even	6	1
525.2.t.c		2	315.u	even	6	1
525.2.t.c		2	315.bn	odd	6	1
567.2.i.b		2	1.a	even	1	1	trivial
567.2.i.b		2	3.b	odd	2	1	CM
567.2.i.b		2	63.i	even	6	1	inner
567.2.i.b		2	63.t	odd	6	1	inner
567.2.s.a		2	7.d	odd	6	1
567.2.s.a		2	9.c	even	3	1
567.2.s.a		2	9.d	odd	6	1
567.2.s.a		2	21.g	even	6	1
2352.2.k.c		2	252.r	odd	6	1
2352.2.k.c		2	252.u	odd	6	1
2352.2.k.c		2	252.bb	even	6	1
2352.2.k.c		2	252.bj	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2} \)

\( T_{11} \)

\( T_{13}^{2} - 3T_{13} + 3 \)

Hecke characteristic polynomials

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