Newform orbit 640.2.o.f

• Label: 640.2.o.f
• Level: $640$
• Weight: $2$
• Character orbit: 640.o
• Analytic conductor: $5.110$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$5.11042572936$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (-i + 2) * q^5 + 3*i * q^9 + (5*i + 5) * q^13 + (-3*i - 3) * q^17 + (-4*i + 3) * q^25 + 10 * q^29 + (-5*i + 5) * q^37 - 8 * q^41 + (6*i + 3) * q^45 + 7*i * q^49 + (-5*i - 5) * q^53 + 10*i * q^61 + (5*i + 15) * q^65 + (-11*i + 11) * q^73 - 9 * q^81 + (-3*i - 9) * q^85 + 16*i * q^89 + (-13*i - 13) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{5} + 10 q^{13} - 6 q^{17} + 6 q^{25} + 20 q^{29} + 10 q^{37} - 16 q^{41} + 6 q^{45} - 10 q^{53} + 30 q^{65} + 22 q^{73} - 18 q^{81} - 18 q^{85} - 26 q^{97}+O(q^{100})$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

63.1

	−	1.00000i
		1.00000i

0

0

0

2.00000

+

1.00000i

0

0

0

−

3.00000i

0

447.1

0

0

0

2.00000

−

1.00000i

0

0

0

3.00000i

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	640.2.o.f	yes	2
4.b	odd	2	1	CM	640.2.o.f	yes	2
5.c	odd	4	1		640.2.o.c	✓	2
8.b	even	2	1		640.2.o.c	✓	2
8.d	odd	2	1		640.2.o.c	✓	2
16.e	even	4	1		1280.2.n.e		2
16.e	even	4	1		1280.2.n.h		2
16.f	odd	4	1		1280.2.n.e		2
16.f	odd	4	1		1280.2.n.h		2
20.e	even	4	1		640.2.o.c	✓	2
40.i	odd	4	1	inner	640.2.o.f	yes	2
40.k	even	4	1	inner	640.2.o.f	yes	2
80.i	odd	4	1		1280.2.n.e		2
80.j	even	4	1		1280.2.n.h		2
80.s	even	4	1		1280.2.n.e		2
80.t	odd	4	1		1280.2.n.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
640.2.o.c	✓	2	5.c	odd	4	1
640.2.o.c	✓	2	8.b	even	2	1
640.2.o.c	✓	2	8.d	odd	2	1
640.2.o.c	✓	2	20.e	even	4	1
640.2.o.f	yes	2	1.a	even	1	1	trivial
640.2.o.f	yes	2	4.b	odd	2	1	CM
640.2.o.f	yes	2	40.i	odd	4	1	inner
640.2.o.f	yes	2	40.k	even	4	1	inner
1280.2.n.e		2	16.e	even	4	1
1280.2.n.e		2	16.f	odd	4	1
1280.2.n.e		2	80.i	odd	4	1
1280.2.n.e		2	80.s	even	4	1
1280.2.n.h		2	16.e	even	4	1
1280.2.n.h		2	16.f	odd	4	1
1280.2.n.h		2	80.j	even	4	1
1280.2.n.h		2	80.t	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(640, [\chi])$:

$T_{3}$

$T_{7}$

$T_{13}^{2} - 10T_{13} + 50$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} - 4T + 5$
$7$	$T^{2}$
$11$	$T^{2}$