Newform orbit 2500.1.b.a

• Label: 2500.1.b.a
• Level: $2500$
• Weight: $1$
• Character orbit: 2500.b
• Self dual: yes
• Analytic conductor: $1.248$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{5}$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2500 = 2^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2500.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.24766253158\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
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 100)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.6250000.1

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times\).

\(n\)	\(1251\)	\(1877\)
\(\chi(n)\)	\(-1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1251.1

1.61803
−0.618034

−1.00000

0

1.00000

0

0

0

−1.00000

1.00000

0

1251.2

−1.00000

0

1.00000

0

0

0

−1.00000

1.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2500.1.b.a		2
4.b	odd	2	1	CM	2500.1.b.a		2
5.b	even	2	1		2500.1.b.b		2
5.c	odd	4	2		2500.1.d.a		4
20.d	odd	2	1		2500.1.b.b		2
20.e	even	4	2		2500.1.d.a		4
25.d	even	5	2		500.1.j.a		4
25.d	even	5	2		2500.1.j.b		4
25.e	even	10	2		100.1.j.a	✓	4
25.e	even	10	2		2500.1.j.a		4
25.f	odd	20	4		500.1.h.a		8
25.f	odd	20	4		2500.1.h.e		8
75.h	odd	10	2		900.1.x.a		4
100.h	odd	10	2		100.1.j.a	✓	4
100.h	odd	10	2		2500.1.j.a		4
100.j	odd	10	2		500.1.j.a		4
100.j	odd	10	2		2500.1.j.b		4
100.l	even	20	4		500.1.h.a		8
100.l	even	20	4		2500.1.h.e		8
200.o	even	10	2		1600.1.bh.a		4
200.s	odd	10	2		1600.1.bh.a		4
300.r	even	10	2		900.1.x.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
100.1.j.a	✓	4	25.e	even	10	2
100.1.j.a	✓	4	100.h	odd	10	2
500.1.h.a		8	25.f	odd	20	4
500.1.h.a		8	100.l	even	20	4
500.1.j.a		4	25.d	even	5	2
500.1.j.a		4	100.j	odd	10	2
900.1.x.a		4	75.h	odd	10	2
900.1.x.a		4	300.r	even	10	2
1600.1.bh.a		4	200.o	even	10	2
1600.1.bh.a		4	200.s	odd	10	2
2500.1.b.a		2	1.a	even	1	1	trivial
2500.1.b.a		2	4.b	odd	2	1	CM
2500.1.b.b		2	5.b	even	2	1
2500.1.b.b		2	20.d	odd	2	1
2500.1.d.a		4	5.c	odd	4	2
2500.1.d.a		4	20.e	even	4	2
2500.1.h.e		8	25.f	odd	20	4
2500.1.h.e		8	100.l	even	20	4
2500.1.j.a		4	25.e	even	10	2
2500.1.j.a		4	100.h	odd	10	2
2500.1.j.b		4	25.d	even	5	2
2500.1.j.b		4	100.j	odd	10	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(2500, [\chi])\).

Hecke characteristic polynomials

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