Newform orbit 847.1.d.b

• Label: 847.1.d.b
• Level: $847$
• Weight: $1$
• Character orbit: 847.d
• Self dual: yes
• Analytic conductor: $0.423$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{5}$
• CM discriminant: -7
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	847.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.422708065700\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 77)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.717409.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta + 1) q^{2} + ( - \beta + 1) q^{4} - q^{7} + q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

727.1

1.61803
−0.618034

−0.618034

0

−0.618034

0

0

−1.00000

1.00000

1.00000

0

727.2

1.61803

0

1.61803

0

0

−1.00000

1.00000

1.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.1.d.b		2
7.b	odd	2	1	CM	847.1.d.b		2
11.b	odd	2	1		847.1.d.a		2
11.c	even	5	2		847.1.j.a		4
11.c	even	5	2		847.1.j.b		4
11.d	odd	10	2		77.1.j.a	✓	4
11.d	odd	10	2		847.1.j.c		4
33.f	even	10	2		693.1.br.a		4
44.g	even	10	2		1232.1.cd.a		4
55.h	odd	10	2		1925.1.bn.a		4
55.l	even	20	4		1925.1.cb.a		8
77.b	even	2	1		847.1.d.a		2
77.j	odd	10	2		847.1.j.a		4
77.j	odd	10	2		847.1.j.b		4
77.l	even	10	2		77.1.j.a	✓	4
77.l	even	10	2		847.1.j.c		4
77.n	even	30	4		539.1.u.a		8
77.o	odd	30	4		539.1.u.a		8
231.r	odd	10	2		693.1.br.a		4
308.s	odd	10	2		1232.1.cd.a		4
385.v	even	10	2		1925.1.bn.a		4
385.bi	odd	20	4		1925.1.cb.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.1.j.a	✓	4	11.d	odd	10	2
77.1.j.a	✓	4	77.l	even	10	2
539.1.u.a		8	77.n	even	30	4
539.1.u.a		8	77.o	odd	30	4
693.1.br.a		4	33.f	even	10	2
693.1.br.a		4	231.r	odd	10	2
847.1.d.a		2	11.b	odd	2	1
847.1.d.a		2	77.b	even	2	1
847.1.d.b		2	1.a	even	1	1	trivial
847.1.d.b		2	7.b	odd	2	1	CM
847.1.j.a		4	11.c	even	5	2
847.1.j.a		4	77.j	odd	10	2
847.1.j.b		4	11.c	even	5	2
847.1.j.b		4	77.j	odd	10	2
847.1.j.c		4	11.d	odd	10	2
847.1.j.c		4	77.l	even	10	2
1232.1.cd.a		4	44.g	even	10	2
1232.1.cd.a		4	308.s	odd	10	2
1925.1.bn.a		4	55.h	odd	10	2
1925.1.bn.a		4	385.v	even	10	2
1925.1.cb.a		8	55.l	even	20	4
1925.1.cb.a		8	385.bi	odd	20	4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(847, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} \)