LMFDB - Newform orbit 880.1.i.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,1,Mod(769,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.769"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	880.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$0.439177211117$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 220)
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.242000.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + ( - \zeta_{6}^{2} - \zeta_{6}) q^{3} - \zeta_{6}^{2} q^{5} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{9} + q^{11} + ( - \zeta_{6} - 1) q^{15} + (\zeta_{6}^{2} + \zeta_{6}) q^{23} - \zeta_{6} q^{25} + (\zeta_{6}^{2} + \zeta_{6}) q^{27} + \cdots + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{99} +O(q^{100}) $ q + (-z^2 - z) * q^3 - z^2 * q^5 + (z^2 - z - 1) * q^9 + q^11 + (-z - 1) * q^15 + (z^2 + z) * q^23 - z * q^25 + (z^2 + z) * q^27 - q^31 + (-z^2 - z) * q^33 + (z^2 + z) * q^37 + (z^2 + z - 1) * q^45 - q^49 - z^2 * q^55 + q^59 + (-z^2 - z) * q^67 + (-z^2 + z + 2) * q^69 + q^71 + (z^2 - 1) * q^75 + q^81 + q^89 + (z^2 + z) * q^93 + (-z^2 - z) * q^97 + (z^2 - z - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - 4 q^{9} + 2 q^{11} - 3 q^{15} - q^{25} - 2 q^{31} - 2 q^{45} - 2 q^{49} + q^{55} + 2 q^{59} + 6 q^{69} + 2 q^{71} - 3 q^{75} + 2 q^{81} + 2 q^{89} - 4 q^{99}+O(q^{100}) $ 2 * q + q^5 - 4 * q^9 + 2 * q^11 - 3 * q^15 - q^25 - 2 * q^31 - 2 * q^45 - 2 * q^49 + q^55 + 2 * q^59 + 6 * q^69 + 2 * q^71 - 3 * q^75 + 2 * q^81 + 2 * q^89 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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} $

769.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−

1.73205i

0.500000

−

0.866025i

−2.00000

769.2

1.73205i

0.500000

0.866025i

−2.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
5.b	even	2	1	inner
55.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	880.1.i.b		2
4.b	odd	2	1		220.1.e.a	✓	2
5.b	even	2	1	inner	880.1.i.b		2
8.b	even	2	1		3520.1.i.c		2
8.d	odd	2	1		3520.1.i.d		2
11.b	odd	2	1	CM	880.1.i.b		2
12.b	even	2	1		1980.1.p.a		2
20.d	odd	2	1		220.1.e.a	✓	2
20.e	even	4	2		1100.1.f.b		2
40.e	odd	2	1		3520.1.i.d		2
40.f	even	2	1		3520.1.i.c		2
44.c	even	2	1		220.1.e.a	✓	2
44.g	even	10	4		2420.1.q.a		8
44.h	odd	10	4		2420.1.q.a		8
55.d	odd	2	1	inner	880.1.i.b		2
60.h	even	2	1		1980.1.p.a		2
88.b	odd	2	1		3520.1.i.c		2
88.g	even	2	1		3520.1.i.d		2
132.d	odd	2	1		1980.1.p.a		2
220.g	even	2	1		220.1.e.a	✓	2
220.i	odd	4	2		1100.1.f.b		2
220.n	odd	10	4		2420.1.q.a		8
220.o	even	10	4		2420.1.q.a		8
440.c	even	2	1		3520.1.i.d		2
440.o	odd	2	1		3520.1.i.c		2
660.g	odd	2	1		1980.1.p.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.1.e.a	✓	2	4.b	odd	2	1
220.1.e.a	✓	2	20.d	odd	2	1
220.1.e.a	✓	2	44.c	even	2	1
220.1.e.a	✓	2	220.g	even	2	1
880.1.i.b		2	1.a	even	1	1	trivial
880.1.i.b		2	5.b	even	2	1	inner
880.1.i.b		2	11.b	odd	2	1	CM
880.1.i.b		2	55.d	odd	2	1	inner
1100.1.f.b		2	20.e	even	4	2
1100.1.f.b		2	220.i	odd	4	2
1980.1.p.a		2	12.b	even	2	1
1980.1.p.a		2	60.h	even	2	1
1980.1.p.a		2	132.d	odd	2	1
1980.1.p.a		2	660.g	odd	2	1
2420.1.q.a		8	44.g	even	10	4
2420.1.q.a		8	44.h	odd	10	4
2420.1.q.a		8	220.n	odd	10	4
2420.1.q.a		8	220.o	even	10	4
3520.1.i.c		2	8.b	even	2	1
3520.1.i.c		2	40.f	even	2	1
3520.1.i.c		2	88.b	odd	2	1
3520.1.i.c		2	440.o	odd	2	1
3520.1.i.d		2	8.d	odd	2	1
3520.1.i.d		2	40.e	odd	2	1
3520.1.i.d		2	88.g	even	2	1
3520.1.i.d		2	440.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} + 3 $ T3^2 + 3 acting on $S_{1}^{\mathrm{new}}(880, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 3 $ T^2 + 3
$5$	$ T^{2} - T + 1 $ T^2 - T + 1
$7$	$ T^{2} $ T^2
$11$	$ (T - 1)^{2} $ (T - 1)^2
$13$	$ T^{2} $ T^2
$17$	$ T^{2} $ T^2
$19$	$ T^{2} $ T^2
$23$	$ T^{2} + 3 $ T^2 + 3
$29$	$ T^{2} $ T^2
$31$	$ (T + 1)^{2} $ (T + 1)^2
$37$	$ T^{2} + 3 $ T^2 + 3
$41$	$ T^{2} $ T^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ T^{2} $ T^2
$59$	$ (T - 1)^{2} $ (T - 1)^2
$61$	$ T^{2} $ T^2
$67$	$ T^{2} + 3 $ T^2 + 3
$71$	$ (T - 1)^{2} $ (T - 1)^2
$73$	$ T^{2} $ T^2
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ (T - 1)^{2} $ (T - 1)^2
$97$	$ T^{2} + 3 $ T^2 + 3
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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	880.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6}^{2} - \zeta_{6}) q^{3} - \zeta_{6}^{2} q^{5} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{9} + q^{11} + ( - \zeta_{6} - 1) q^{15} + (\zeta_{6}^{2} + \zeta_{6}) q^{23} - \zeta_{6} q^{25} + (\zeta_{6}^{2} + \zeta_{6}) q^{27} + \cdots + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{99} +O(q^{100}) \) q + (-z^2 - z) * q^3 - z^2 * q^5 + (z^2 - z - 1) * q^9 + q^11 + (-z - 1) * q^15 + (z^2 + z) * q^23 - z * q^25 + (z^2 + z) * q^27 - q^31 + (-z^2 - z) * q^33 + (z^2 + z) * q^37 + (z^2 + z - 1) * q^45 - q^49 - z^2 * q^55 + q^59 + (-z^2 - z) * q^67 + (-z^2 + z + 2) * q^69 + q^71 + (z^2 - 1) * q^75 + q^81 + q^89 + (z^2 + z) * q^93 + (-z^2 - z) * q^97 + (z^2 - z - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - 4 q^{9} + 2 q^{11} - 3 q^{15} - q^{25} - 2 q^{31} - 2 q^{45} - 2 q^{49} + q^{55} + 2 q^{59} + 6 q^{69} + 2 q^{71} - 3 q^{75} + 2 q^{81} + 2 q^{89} - 4 q^{99}+O(q^{100}) \) 2 * q + q^5 - 4 * q^9 + 2 * q^11 - 3 * q^15 - q^25 - 2 * q^31 - 2 * q^45 - 2 * q^49 + q^55 + 2 * q^59 + 6 * q^69 + 2 * q^71 - 3 * q^75 + 2 * q^81 + 2 * q^89 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more