Newform orbit 500.1.d.b

• Label: 500.1.d.b
• Level: $500$
• Weight: $1$
• Character orbit: 500.d
• Self dual: yes
• Analytic conductor: $0.250$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{5}$
• CM discriminant: -20
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	500.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.249532506317\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.250000.1
Artin image:	$D_5$
Artin field:	Galois closure of 5.1.250000.1

$q$-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.

\(f(q)\)	\(=\)	\( q + q^{2} + (\beta - 1) q^{3} + q^{4} + (\beta - 1) q^{6} - \beta q^{7} + q^{8} + ( - \beta + 1) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} - q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\)	\(251\)	\(377\)
\(\chi(n)\)	\(-1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

499.1

−0.618034
1.61803

1.00000

−1.61803

1.00000

0

−1.61803

0.618034

1.00000

1.61803

0

499.2

1.00000

0.618034

1.00000

0

0.618034

−1.61803

1.00000

−0.618034

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	500.1.d.b		2
4.b	odd	2	1		500.1.d.a		2
5.b	even	2	1		500.1.d.a		2
5.c	odd	4	2		500.1.b.a	✓	4
20.d	odd	2	1	CM	500.1.d.b		2
20.e	even	4	2		500.1.b.a	✓	4
25.d	even	5	2		2500.1.h.a		4
25.d	even	5	2		2500.1.h.b		4
25.e	even	10	2		2500.1.h.c		4
25.e	even	10	2		2500.1.h.d		4
25.f	odd	20	4		2500.1.j.c		8
25.f	odd	20	4		2500.1.j.d		8
100.h	odd	10	2		2500.1.h.a		4
100.h	odd	10	2		2500.1.h.b		4
100.j	odd	10	2		2500.1.h.c		4
100.j	odd	10	2		2500.1.h.d		4
100.l	even	20	4		2500.1.j.c		8
100.l	even	20	4		2500.1.j.d		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
500.1.b.a	✓	4	5.c	odd	4	2
500.1.b.a	✓	4	20.e	even	4	2
500.1.d.a		2	4.b	odd	2	1
500.1.d.a		2	5.b	even	2	1
500.1.d.b		2	1.a	even	1	1	trivial
500.1.d.b		2	20.d	odd	2	1	CM
2500.1.h.a		4	25.d	even	5	2
2500.1.h.a		4	100.h	odd	10	2
2500.1.h.b		4	25.d	even	5	2
2500.1.h.b		4	100.h	odd	10	2
2500.1.h.c		4	25.e	even	10	2
2500.1.h.c		4	100.j	odd	10	2
2500.1.h.d		4	25.e	even	10	2
2500.1.h.d		4	100.j	odd	10	2
2500.1.j.c		8	25.f	odd	20	4
2500.1.j.c		8	100.l	even	20	4
2500.1.j.d		8	25.f	odd	20	4
2500.1.j.d		8	100.l	even	20	4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{1}^{\mathrm{new}}(500, [\chi])\).

Hecke characteristic polynomials

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