Newform orbit 640.1.m.b

• Label: 640.1.m.b
• Level: $640$
• Weight: $1$
• Character orbit: 640.m
• Analytic conductor: $0.319$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$640 = 2^{7} \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	640.m (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.319401608085$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.32000.1
Artin image:	$C_4\wr C_2$
Artin field:	Galois closure of 8.0.131072000.1

$q$-expansion

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	q + z * q^5 - z * q^9 + (z + 1) * q^13 + (z + 1) * q^17 - q^25 - 2 * q^29 + (-z + 1) * q^37 + q^45 - z * q^49 + (-z - 1) * q^53 - 2*z * q^61 + (z - 1) * q^65 + (z - 1) * q^73 - q^81 + (z - 1) * q^85 + (-z - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{13} + 2 q^{17} - 2 q^{25} - 4 q^{29} + 2 q^{37} + 2 q^{45} - 2 q^{53} - 2 q^{65} - 2 q^{73} - 2 q^{81} - 2 q^{85} - 2 q^{97}+O(q^{100})$

Character values

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

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

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

193.1

	−	1.00000i
		1.00000i

0

0

0

−

1.00000i

0

0

0

1.00000i

0

577.1

0

0

0

1.00000i

0

0

0

−

1.00000i

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.i	odd	4	1	inner
40.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	640.1.m.b	yes	2
4.b	odd	2	1	CM	640.1.m.b	yes	2
5.b	even	2	1		3200.1.m.a		2
5.c	odd	4	1		640.1.m.a	✓	2
5.c	odd	4	1		3200.1.m.b		2
8.b	even	2	1		640.1.m.a	✓	2
8.d	odd	2	1		640.1.m.a	✓	2
16.e	even	4	1		1280.1.p.a		2
16.e	even	4	1		1280.1.p.b		2
16.f	odd	4	1		1280.1.p.a		2
16.f	odd	4	1		1280.1.p.b		2
20.d	odd	2	1		3200.1.m.a		2
20.e	even	4	1		640.1.m.a	✓	2
20.e	even	4	1		3200.1.m.b		2
40.e	odd	2	1		3200.1.m.b		2
40.f	even	2	1		3200.1.m.b		2
40.i	odd	4	1	inner	640.1.m.b	yes	2
40.i	odd	4	1		3200.1.m.a		2
40.k	even	4	1	inner	640.1.m.b	yes	2
40.k	even	4	1		3200.1.m.a		2
80.i	odd	4	1		1280.1.p.b		2
80.j	even	4	1		1280.1.p.a		2
80.s	even	4	1		1280.1.p.b		2
80.t	odd	4	1		1280.1.p.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
640.1.m.a	✓	2	5.c	odd	4	1
640.1.m.a	✓	2	8.b	even	2	1
640.1.m.a	✓	2	8.d	odd	2	1
640.1.m.a	✓	2	20.e	even	4	1
640.1.m.b	yes	2	1.a	even	1	1	trivial
640.1.m.b	yes	2	4.b	odd	2	1	CM
640.1.m.b	yes	2	40.i	odd	4	1	inner
640.1.m.b	yes	2	40.k	even	4	1	inner
1280.1.p.a		2	16.e	even	4	1
1280.1.p.a		2	16.f	odd	4	1
1280.1.p.a		2	80.j	even	4	1
1280.1.p.a		2	80.t	odd	4	1
1280.1.p.b		2	16.e	even	4	1
1280.1.p.b		2	16.f	odd	4	1
1280.1.p.b		2	80.i	odd	4	1
1280.1.p.b		2	80.s	even	4	1
3200.1.m.a		2	5.b	even	2	1
3200.1.m.a		2	20.d	odd	2	1
3200.1.m.a		2	40.i	odd	4	1
3200.1.m.a		2	40.k	even	4	1
3200.1.m.b		2	5.c	odd	4	1
3200.1.m.b		2	20.e	even	4	1
3200.1.m.b		2	40.e	odd	2	1
3200.1.m.b		2	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{13}^{2} - 2T_{13} + 2$ acting on $S_{1}^{\mathrm{new}}(640, [\chi])$.

Hecke characteristic polynomials

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