Newform orbit 1000.1.e.a

• Label: 1000.1.e.a
• Level: $1000$
• Weight: $1$
• Character orbit: 1000.e
• Self dual: yes
• Analytic conductor: $0.499$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{5}$
• CM discriminant: -40
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1000.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.499065012633\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.1000000.1

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\)	\(377\)	\(501\)	\(751\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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

1.61803
−0.618034

−1.00000

0

1.00000

0

0

−0.618034

−1.00000

1.00000

0

499.2

−1.00000

0

1.00000

0

0

1.61803

−1.00000

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by \(\Q(\sqrt{-10}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1000.1.e.a		2
4.b	odd	2	1		4000.1.e.a		2
5.b	even	2	1		1000.1.e.b		2
5.c	odd	4	2		1000.1.g.a	✓	4
8.b	even	2	1		4000.1.e.b		2
8.d	odd	2	1		1000.1.e.b		2
20.d	odd	2	1		4000.1.e.b		2
20.e	even	4	2		4000.1.g.a		4
40.e	odd	2	1	CM	1000.1.e.a		2
40.f	even	2	1		4000.1.e.a		2
40.i	odd	4	2		4000.1.g.a		4
40.k	even	4	2		1000.1.g.a	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1000.1.e.a		2	1.a	even	1	1	trivial
1000.1.e.a		2	40.e	odd	2	1	CM
1000.1.e.b		2	5.b	even	2	1
1000.1.e.b		2	8.d	odd	2	1
1000.1.g.a	✓	4	5.c	odd	4	2
1000.1.g.a	✓	4	40.k	even	4	2
4000.1.e.a		2	4.b	odd	2	1
4000.1.e.a		2	40.f	even	2	1
4000.1.e.b		2	8.b	even	2	1
4000.1.e.b		2	20.d	odd	2	1
4000.1.g.a		4	20.e	even	4	2
4000.1.g.a		4	40.i	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - T_{7} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1000, [\chi])\).

Hecke characteristic polynomials

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