LMFDB - Newform orbit 640.2.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(449,640)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("640.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.11042572936$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta - 1) q^{5} - 3 q^{9}+O(q^{10}) $ q + (b - 1) * q^5 - 3 * q^9	$ q + (\beta - 1) q^{5} - 3 q^{9} - 6 q^{13} - 4 \beta q^{17} + ( - 2 \beta - 3) q^{25} + 2 \beta q^{29} - 2 q^{37} - 10 q^{41} + ( - 3 \beta + 3) q^{45} + 7 q^{49} - 14 q^{53} + 6 \beta q^{61} + ( - 6 \beta + 6) q^{65} + 8 \beta q^{73} + 9 q^{81} + (4 \beta + 16) q^{85} - 10 q^{89} + 4 \beta q^{97} +O(q^{100}) $ q + (b - 1) * q^5 - 3 * q^9 - 6 * q^13 - 4b q^17 + (-2b - 3) q^25 + 2b q^29 - 2 * q^37 - 10 * q^41 + (-3b + 3) q^45 + 7 * q^49 - 14 * q^53 + 6b q^61 + (-6b + 6) q^65 + 8b q^73 + 9 * q^81 + (4b + 16) q^85 - 10 * q^89 + 4b q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 6 * q^9	$ 2 q - 2 q^{5} - 6 q^{9} - 12 q^{13} - 6 q^{25} - 4 q^{37} - 20 q^{41} + 6 q^{45} + 14 q^{49} - 28 q^{53} + 12 q^{65} + 18 q^{81} + 32 q^{85} - 20 q^{89}+O(q^{100}) $ 2 * q - 2 * q^5 - 6 * q^9 - 12 * q^13 - 6 * q^25 - 4 * q^37 - 20 * q^41 + 6 * q^45 + 14 * q^49 - 28 * q^53 + 12 * q^65 + 18 * q^81 + 32 * q^85 - 20 * q^89

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

449.1

	−	1.00000i
		1.00000i

−1.00000

−

2.00000i

−3.00000

449.2

−1.00000

2.00000i

−3.00000

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	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	640.2.f.a	✓	2
4.b	odd	2	1	CM	640.2.f.a	✓	2
5.b	even	2	1		640.2.f.b	yes	2
5.c	odd	4	1		3200.2.d.d		2
5.c	odd	4	1		3200.2.d.f		2
8.b	even	2	1		640.2.f.b	yes	2
8.d	odd	2	1		640.2.f.b	yes	2
16.e	even	4	1		1280.2.c.a		2
16.e	even	4	1		1280.2.c.d		2
16.f	odd	4	1		1280.2.c.a		2
16.f	odd	4	1		1280.2.c.d		2
20.d	odd	2	1		640.2.f.b	yes	2
20.e	even	4	1		3200.2.d.d		2
20.e	even	4	1		3200.2.d.f		2
40.e	odd	2	1	inner	640.2.f.a	✓	2
40.f	even	2	1	inner	640.2.f.a	✓	2
40.i	odd	4	1		3200.2.d.d		2
40.i	odd	4	1		3200.2.d.f		2
40.k	even	4	1		3200.2.d.d		2
40.k	even	4	1		3200.2.d.f		2
80.i	odd	4	1		6400.2.a.n		1
80.i	odd	4	1		6400.2.a.o		1
80.j	even	4	1		6400.2.a.j		1
80.j	even	4	1		6400.2.a.k		1
80.k	odd	4	1		1280.2.c.a		2
80.k	odd	4	1		1280.2.c.d		2
80.q	even	4	1		1280.2.c.a		2
80.q	even	4	1		1280.2.c.d		2
80.s	even	4	1		6400.2.a.n		1
80.s	even	4	1		6400.2.a.o		1
80.t	odd	4	1		6400.2.a.j		1
80.t	odd	4	1		6400.2.a.k		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
640.2.f.a	✓	2	1.a	even	1	1	trivial
640.2.f.a	✓	2	4.b	odd	2	1	CM
640.2.f.a	✓	2	40.e	odd	2	1	inner
640.2.f.a	✓	2	40.f	even	2	1	inner
640.2.f.b	yes	2	5.b	even	2	1
640.2.f.b	yes	2	8.b	even	2	1
640.2.f.b	yes	2	8.d	odd	2	1
640.2.f.b	yes	2	20.d	odd	2	1
1280.2.c.a		2	16.e	even	4	1
1280.2.c.a		2	16.f	odd	4	1
1280.2.c.a		2	80.k	odd	4	1
1280.2.c.a		2	80.q	even	4	1
1280.2.c.d		2	16.e	even	4	1
1280.2.c.d		2	16.f	odd	4	1
1280.2.c.d		2	80.k	odd	4	1
1280.2.c.d		2	80.q	even	4	1
3200.2.d.d		2	5.c	odd	4	1
3200.2.d.d		2	20.e	even	4	1
3200.2.d.d		2	40.i	odd	4	1
3200.2.d.d		2	40.k	even	4	1
3200.2.d.f		2	5.c	odd	4	1
3200.2.d.f		2	20.e	even	4	1
3200.2.d.f		2	40.i	odd	4	1
3200.2.d.f		2	40.k	even	4	1
6400.2.a.j		1	80.j	even	4	1
6400.2.a.j		1	80.t	odd	4	1
6400.2.a.k		1	80.j	even	4	1
6400.2.a.k		1	80.t	odd	4	1
6400.2.a.n		1	80.i	odd	4	1
6400.2.a.n		1	80.s	even	4	1
6400.2.a.o		1	80.i	odd	4	1
6400.2.a.o		1	80.s	even	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} + 6 $

$ T_{37} + 2 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 2T + 5 $ T^2 + 2*T + 5
$7$	$ T^{2} $ T^2
$11$	$ T^{2} $ T^2
$13$	$ (T + 6)^{2} $ (T + 6)^2
$17$	$ T^{2} + 64 $ T^2 + 64
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} + 16 $ T^2 + 16
$31$	$ T^{2} $ T^2
$37$	$ (T + 2)^{2} $ (T + 2)^2
$41$	$ (T + 10)^{2} $ (T + 10)^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ (T + 14)^{2} $ (T + 14)^2
$59$	$ T^{2} $ T^2
$61$	$ T^{2} + 144 $ T^2 + 144
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ T^{2} + 256 $ T^2 + 256
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ (T + 10)^{2} $ (T + 10)^2
$97$	$ T^{2} + 64 $ T^2 + 64
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\( q + (\beta - 1) q^{5} - 3 q^{9}+O(q^{10}) \) q + (b - 1) * q^5 - 3 * q^9	\( q + (\beta - 1) q^{5} - 3 q^{9} - 6 q^{13} - 4 \beta q^{17} + ( - 2 \beta - 3) q^{25} + 2 \beta q^{29} - 2 q^{37} - 10 q^{41} + ( - 3 \beta + 3) q^{45} + 7 q^{49} - 14 q^{53} + 6 \beta q^{61} + ( - 6 \beta + 6) q^{65} + 8 \beta q^{73} + 9 q^{81} + (4 \beta + 16) q^{85} - 10 q^{89} + 4 \beta q^{97} +O(q^{100}) \) q + (b - 1) * q^5 - 3 * q^9 - 6 * q^13 - 4b q^17 + (-2b - 3) q^25 + 2b q^29 - 2 * q^37 - 10 * q^41 + (-3b + 3) q^45 + 7 * q^49 - 14 * q^53 + 6b q^61 + (-6b + 6) q^65 + 8b q^73 + 9 * q^81 + (4b + 16) q^85 - 10 * q^89 + 4b q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 6 * q^9	\( 2 q - 2 q^{5} - 6 q^{9} - 12 q^{13} - 6 q^{25} - 4 q^{37} - 20 q^{41} + 6 q^{45} + 14 q^{49} - 28 q^{53} + 12 q^{65} + 18 q^{81} + 32 q^{85} - 20 q^{89}+O(q^{100}) \) 2 * q - 2 * q^5 - 6 * q^9 - 12 * q^13 - 6 * q^25 - 4 * q^37 - 20 * q^41 + 6 * q^45 + 14 * q^49 - 28 * q^53 + 12 * q^65 + 18 * q^81 + 32 * q^85 - 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more