Newform orbit 640.1.e.b

• Label: 640.1.e.b
• Level: $640$
• Weight: $1$
• Character orbit: 640.e
• Self dual: yes
• Analytic conductor: $0.319$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -4, -40, 40
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	640.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.319401608085\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(i, \sqrt{10})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.2560.1
Stark unit:	Root of $x^{4} - 100x^{3} - 58x^{2} - 100x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{5} + q^{9} - 2 q^{13} + q^{25} + 2 q^{37} - 2 q^{41} + q^{45} - q^{49} - 2 q^{53} - 2 q^{65} + q^{81} - 2 q^{89}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\)	\(257\)	\(261\)	\(511\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

319.1

0

0

0

0

1.00000

0

0

0

1.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	640.1.e.b	yes	1
4.b	odd	2	1	CM	640.1.e.b	yes	1
5.b	even	2	1		640.1.e.a	✓	1
5.c	odd	4	2		3200.1.g.b		2
8.b	even	2	1		640.1.e.a	✓	1
8.d	odd	2	1		640.1.e.a	✓	1
16.e	even	4	2		1280.1.h.b		2
16.f	odd	4	2		1280.1.h.b		2
20.d	odd	2	1		640.1.e.a	✓	1
20.e	even	4	2		3200.1.g.b		2
40.e	odd	2	1	CM	640.1.e.b	yes	1
40.f	even	2	1	RM	640.1.e.b	yes	1
40.i	odd	4	2		3200.1.g.b		2
40.k	even	4	2		3200.1.g.b		2
80.k	odd	4	2		1280.1.h.b		2
80.q	even	4	2		1280.1.h.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
640.1.e.a	✓	1	5.b	even	2	1
640.1.e.a	✓	1	8.b	even	2	1
640.1.e.a	✓	1	8.d	odd	2	1
640.1.e.a	✓	1	20.d	odd	2	1
640.1.e.b	yes	1	1.a	even	1	1	trivial
640.1.e.b	yes	1	4.b	odd	2	1	CM
640.1.e.b	yes	1	40.e	odd	2	1	CM
640.1.e.b	yes	1	40.f	even	2	1	RM
1280.1.h.b		2	16.e	even	4	2
1280.1.h.b		2	16.f	odd	4	2
1280.1.h.b		2	80.k	odd	4	2
1280.1.h.b		2	80.q	even	4	2
3200.1.g.b		2	5.c	odd	4	2
3200.1.g.b		2	20.e	even	4	2
3200.1.g.b		2	40.i	odd	4	2
3200.1.g.b		2	40.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(640, [\chi])\):

\( T_{3} \)

\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T - 1 \)
$7$	\( T \)
$11$	\( T \)