LMFDB - Newform orbit 4000.1.bf.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,1,Mod(799,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 0, 7]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("4000.799");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	4000.bf (of order $10$, degree $4$, not 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:	$1.99626005053$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 800)
Projective image:	$A_{5}$
Projective field:	Galois closure of 5.1.25000000.2

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$\zeta_{20}^{2}$	$1$	$-1$

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} $

799.1

−0.587785	+	0.809017i
0.587785	−	0.809017i
0.951057	−	0.309017i
−0.951057	+	0.309017i
0.951057	+	0.309017i
−0.951057	−	0.309017i
−0.587785	−	0.809017i
0.587785	+	0.809017i

−0.309017

−

0.951057i

−1.61803

799.2

−0.309017

−

0.951057i

−1.61803

1599.1

0.809017

−

0.587785i

0.618034

1599.2

0.809017

−

0.587785i

0.618034

2399.1

0.809017

0.587785i

0.618034

2399.2

0.809017

0.587785i

0.618034

3199.1

−0.309017

0.951057i

−1.61803

3199.2

−0.309017

0.951057i

−1.61803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	inner
25.d	even	5	1	inner
100.h	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4000.1.bf.b		8
4.b	odd	2	1		4000.1.bf.a		8
5.b	even	2	1		4000.1.bf.a		8
5.c	odd	4	1		800.1.bh.a	✓	8
5.c	odd	4	1		4000.1.bh.a		8
20.d	odd	2	1	inner	4000.1.bf.b		8
20.e	even	4	1		800.1.bh.a	✓	8
20.e	even	4	1		4000.1.bh.a		8
25.d	even	5	1	inner	4000.1.bf.b		8
25.e	even	10	1		4000.1.bf.a		8
25.f	odd	20	1		800.1.bh.a	✓	8
25.f	odd	20	1		4000.1.bh.a		8
40.i	odd	4	1		1600.1.bh.b		8
40.k	even	4	1		1600.1.bh.b		8
100.h	odd	10	1	inner	4000.1.bf.b		8
100.j	odd	10	1		4000.1.bf.a		8
100.l	even	20	1		800.1.bh.a	✓	8
100.l	even	20	1		4000.1.bh.a		8
200.v	even	20	1		1600.1.bh.b		8
200.x	odd	20	1		1600.1.bh.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
800.1.bh.a	✓	8	5.c	odd	4	1
800.1.bh.a	✓	8	20.e	even	4	1
800.1.bh.a	✓	8	25.f	odd	20	1
800.1.bh.a	✓	8	100.l	even	20	1
1600.1.bh.b		8	40.i	odd	4	1
1600.1.bh.b		8	40.k	even	4	1
1600.1.bh.b		8	200.v	even	20	1
1600.1.bh.b		8	200.x	odd	20	1
4000.1.bf.a		8	4.b	odd	2	1
4000.1.bf.a		8	5.b	even	2	1
4000.1.bf.a		8	25.e	even	10	1
4000.1.bf.a		8	100.j	odd	10	1
4000.1.bf.b		8	1.a	even	1	1	trivial
4000.1.bf.b		8	20.d	odd	2	1	inner
4000.1.bf.b		8	25.d	even	5	1	inner
4000.1.bf.b		8	100.h	odd	10	1	inner
4000.1.bh.a		8	5.c	odd	4	1
4000.1.bh.a		8	20.e	even	4	1
4000.1.bh.a		8	25.f	odd	20	1
4000.1.bh.a		8	100.l	even	20	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1 $ T3^4 - T3^3 + T3^2 - T3 + 1 acting on $S_{1}^{\mathrm{new}}(4000, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$5$	$ T^{8} $ T^8
$7$	$ (T^{2} + T - 1)^{4} $ (T^2 + T - 1)^4
$11$	$ T^{8} $ T^8
$13$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$17$	$ T^{8} $ T^8
$19$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$23$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$29$	$ (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$31$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$37$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$41$	$ T^{8} $ T^8
$43$	$ (T^{2} - T - 1)^{4} $ (T^2 - T - 1)^4
$47$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$53$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$59$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 6T^4 + T^2 + 1
$61$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$67$	$ T^{8} $ T^8
$71$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 6T^4 + T^2 + 1
$73$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$79$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$83$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$89$	$ T^{8} $ T^8
$97$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 6T^4 + T^2 + 1
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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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	4000.bf (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{20}^{2} q^{3} + ( - \zeta_{20}^{6} + \zeta_{20}^{4}) q^{7}+O(q^{10}) \) q + z^2 * q^3 + (-z^6 + z^4) * q^7	\( q + \zeta_{20}^{2} q^{3} + ( - \zeta_{20}^{6} + \zeta_{20}^{4}) q^{7} + (\zeta_{20}^{5} - \zeta_{20}^{3}) q^{13} + (\zeta_{20}^{5} + \zeta_{20}) q^{19} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{21} + \zeta_{20}^{6} q^{23} - \zeta_{20}^{6} q^{27} + (\zeta_{20}^{4} + 1) q^{29} - \zeta_{20}^{3} q^{31} + \zeta_{20}^{9} q^{37} + (\zeta_{20}^{7} - \zeta_{20}^{5}) q^{39} + ( - \zeta_{20}^{8} + \zeta_{20}^{2}) q^{43} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{47} + (\zeta_{20}^{8} - \zeta_{20}^{2} + 1) q^{49} + \zeta_{20}^{7} q^{53} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{57} + ( - \zeta_{20}^{7} + \zeta_{20}) q^{59} - \zeta_{20}^{6} q^{61} + \zeta_{20}^{8} q^{69} + ( - \zeta_{20}^{3} + \zeta_{20}) q^{71} + \zeta_{20} q^{73} + ( - \zeta_{20}^{9} - \zeta_{20}^{5}) q^{79} - \zeta_{20}^{8} q^{81} + \zeta_{20}^{8} q^{83} + (\zeta_{20}^{6} + \zeta_{20}^{2}) q^{87} + (2 \zeta_{20}^{9} - \zeta_{20}^{7} + \zeta_{20}) q^{91} - \zeta_{20}^{5} q^{93} + ( - \zeta_{20}^{3} + \zeta_{20}) q^{97} +O(q^{100}) \) q + z^2 * q^3 + (-z^6 + z^4) * q^7 + (z^5 - z^3) * q^13 + (z^5 + z) * q^19 + (-z^8 + z^6) * q^21 + z^6 * q^23 - z^6 * q^27 + (z^4 + 1) * q^29 - z^3 * q^31 + z^9 * q^37 + (z^7 - z^5) * q^39 + (-z^8 + z^2) * q^43 + (-z^8 + z^6) * q^47 + (z^8 - z^2 + 1) * q^49 + z^7 * q^53 + (z^7 + z^3) * q^57 + (-z^7 + z) * q^59 - z^6 * q^61 + z^8 * q^69 + (-z^3 + z) * q^71 + z * q^73 + (-z^9 - z^5) * q^79 - z^8 * q^81 + z^8 * q^83 + (z^6 + z^2) * q^87 + (2z^9 - z^7 + z) q^91 - z^5 * q^93 + (-z^3 + z) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 4 q^{7}+O(q^{10}) \) 8 * q + 2 * q^3 - 4 * q^7	\( 8 q + 2 q^{3} - 4 q^{7} + 4 q^{21} + 2 q^{23} - 2 q^{27} + 6 q^{29} + 4 q^{43} + 4 q^{47} + 4 q^{49} - 2 q^{61} - 2 q^{69} + 2 q^{81} - 2 q^{83} + 4 q^{87}+O(q^{100}) \) 8 * q + 2 * q^3 - 4 * q^7 + 4 * q^21 + 2 * q^23 - 2 * q^27 + 6 * q^29 + 4 * q^43 + 4 * q^47 + 4 * q^49 - 2 * q^61 - 2 * q^69 + 2 * q^81 - 2 * q^83 + 4 * q^87

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4000.1.bf.b

Level	$4000$
Weight	$1$
Character orbit	4000.bf
Analytic conductor	$1.996$
Analytic rank	$0$
Dimension	$8$
Projective image	$A_{5}$
CM/RM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.99626005053\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 800)
Projective image:	\(A_{5}\)
Projective field:	Galois closure of 5.1.25000000.2

\(n\)	\(1377\)	\(2501\)	\(2751\)
\(\chi(n)\)	\(\zeta_{20}^{2}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 799.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + T^2 - T + 1)^2
$5$	\( T^{8} \) T^8
$7$	\( (T^{2} + T - 1)^{4} \) (T^2 + T - 1)^4
$11$	\( T^{8} \) T^8
$13$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$17$	\( T^{8} \) T^8
$19$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$23$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + T^2 - T + 1)^2
$29$	\( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$31$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$37$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$41$	\( T^{8} \) T^8
$43$	\( (T^{2} - T - 1)^{4} \) (T^2 - T - 1)^4
$47$	\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$53$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$59$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 6T^4 + T^2 + 1
$61$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$67$	\( T^{8} \) T^8
$71$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 6T^4 + T^2 + 1
$73$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$79$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$83$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$89$	\( T^{8} \) T^8
$97$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 6T^4 + T^2 + 1
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