LMFDB - Newform orbit 2000.1.bp.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2000,1,Mod(79,2000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2000, base_ring=CyclotomicField(50))

chi = DirichletCharacter(H, H._module([25, 0, 41]))

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

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

chi := DirichletCharacter("2000.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2000 = 2^{4} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2000.bp (of order $50$, degree $20$, 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:	$0.998130025266$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{50})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{15} + x^{10} - x^{5} + 1 $ x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{50}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{50} - \cdots)$

$q$-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_{50}^{9} q^{5} + \zeta_{50}^{7} q^{9} + ( - \zeta_{50}^{11} + \zeta_{50}^{3}) q^{13} + ( - \zeta_{50}^{17} - \zeta_{50}^{6}) q^{17} + \zeta_{50}^{18} q^{25} + (\zeta_{50}^{8} + \zeta_{50}^{4}) q^{29}+ \cdots + ( - \zeta_{50}^{23} - \zeta_{50}^{14}) q^{97}+O(q^{100}) $ q + z^9 * q^5 + z^7 * q^9 + (-z^11 + z^3) * q^13 + (-z^17 - z^6) * q^17 + z^18 * q^25 + (z^8 + z^4) * q^29 + (z^24 + z^5) * q^37 + (z^23 + z^21) * q^41 + z^16 * q^45 - z^10 * q^49 + (z^15 + z^2) * q^53 + (z^19 - z^12) * q^61 + (-z^20 + z^12) * q^65 + (-z^22 - z^19) * q^73 + z^14 * q^81 + (-z^15 + z) * q^85 + (-z^21 + z^20) * q^89 + (-z^23 - z^14) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{37} + 5 q^{49} + 5 q^{53} + 5 q^{65} - 5 q^{85} - 5 q^{89}+O(q^{100}) $ 20 * q + 5 * q^37 + 5 * q^49 + 5 * q^53 + 5 * q^65 - 5 * q^85 - 5 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times$.

$n$	$501$	$751$	$1377$
$\chi(n)$	$1$	$-1$	$\zeta_{50}$

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} $

79.1

0.425779	−	0.904827i
−0.0627905	−	0.998027i
−0.0627905	+	0.998027i
−0.728969	+	0.684547i
−0.876307	+	0.481754i
−0.535827	+	0.844328i
0.637424	−	0.770513i
0.992115	−	0.125333i
0.992115	+	0.125333i
0.929776	−	0.368125i
−0.968583	+	0.248690i
−0.876307	−	0.481754i
−0.728969	−	0.684547i
−0.968583	−	0.248690i
0.929776	+	0.368125i
0.425779	+	0.904827i
0.187381	+	0.982287i
0.637424	+	0.770513i
−0.535827	−	0.844328i
0.187381	−	0.982287i

−0.728969

0.684547i

−0.0627905

−

0.998027i

159.1

−0.535827

−

0.844328i

0.425779

0.904827i

239.1

−0.535827

0.844328i

0.425779

−

0.904827i

319.1

−0.876307

0.481754i

−0.535827

−

0.844328i

479.1

0.187381

−

0.982287i

0.929776

−

0.368125i

559.1

0.929776

0.368125i

−0.728969

0.684547i

639.1

−0.0627905

−

0.998027i

0.992115

0.125333i

719.1

0.425779

−

0.904827i

0.637424

−

0.770513i

879.1

0.425779

0.904827i

0.637424

0.770513i

959.1

−0.968583

0.248690i

−0.876307

−

0.481754i

1039.1

0.637424

0.770513i

0.187381

0.982287i

1119.1

0.187381

0.982287i

0.929776

0.368125i

1279.1

−0.876307

−

0.481754i

−0.535827

0.844328i

1359.1

0.637424

−

0.770513i

0.187381

−

0.982287i

1439.1

−0.968583

−

0.248690i

−0.876307

0.481754i

1519.1

−0.728969

−

0.684547i

−0.0627905

0.998027i

1679.1

0.992115

−

0.125333i

−0.968583

−

0.248690i

1759.1

−0.0627905

0.998027i

0.992115

−

0.125333i

1839.1

0.929776

−

0.368125i

−0.728969

−

0.684547i

1919.1

0.992115

0.125333i

−0.968583

0.248690i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
125.h	even	50	1	inner
500.n	odd	50	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2000.1.bp.a	✓	20
4.b	odd	2	1	CM	2000.1.bp.a	✓	20
125.h	even	50	1	inner	2000.1.bp.a	✓	20
500.n	odd	50	1	inner	2000.1.bp.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2000.1.bp.a	✓	20	1.a	even	1	1	trivial
2000.1.bp.a	✓	20	4.b	odd	2	1	CM
2000.1.bp.a	✓	20	125.h	even	50	1	inner
2000.1.bp.a	✓	20	500.n	odd	50	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2000, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ T^{20} $ T^20
$5$	$ T^{20} - T^{15} + \cdots + 1 $ T^20 - T^15 + T^10 - T^5 + 1
$7$	$ T^{20} $ T^20
$11$	$ T^{20} $ T^20
$13$	$ T^{20} + 20 T^{17} + \cdots + 5 $ T^20 + 20T^17 - 5T^16 + 150T^14 - 75T^13 + 25T^12 + 525T^11 - 375T^10 + 250T^9 + 750T^8 - 750T^7 + 650T^6 + 5T^5 + 125T^3 + 25T^2 - 25*T + 5
$17$	$ T^{20} - 5 T^{17} + \cdots + 5 $ T^20 - 5T^17 - 5T^16 + 25T^14 - 75T^13 + 50T^12 - 125T^11 + 250T^10 + 125T^9 + 375T^8 - 475T^7 + 575T^6 - 620T^5 + 625T^4 - 500T^3 + 225T^2 - 50T + 5
$19$	$ T^{20} $ T^20
$23$	$ T^{20} $ T^20
$29$	$ T^{20} + 5 T^{17} + \cdots + 1 $ T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$31$	$ T^{20} $ T^20
$37$	$ T^{20} - 5 T^{19} + \cdots + 5 $ T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5
$41$	$ T^{20} - 5 T^{17} + \cdots + 1 $ T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 - 50T^13 + 45T^12 - 145T^11 + 379T^10 - 525T^9 + 1050T^8 - 2685T^7 + 4160T^6 - 3628T^5 + 1850T^4 - 575T^3 + 110T^2 - 10*T + 1
$43$	$ T^{20} $ T^20
$47$	$ T^{20} $ T^20
$53$	$ T^{20} - 5 T^{19} + \cdots + 5 $ T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5
$59$	$ T^{20} $ T^20
$61$	$ T^{20} + 20 T^{17} + \cdots + 1 $ T^20 + 20T^17 + 5T^16 - 2T^15 + 150T^14 + 75T^13 - 5T^12 + 505T^11 + 379T^10 + 100T^9 + 800T^8 + 640T^7 + 360T^6 + 747T^5 + 600T^4 + 175T^3 + 85T^2 + 15*T + 1
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} - 5 T^{17} + \cdots + 5 $ T^20 - 5T^17 - 5T^16 + 25T^14 + 50T^13 + 25T^12 - 125T^11 - 375T^10 - 375T^9 + 500T^8 + 2525T^7 + 4400T^6 + 4380T^5 + 2750T^4 + 1125T^3 + 300T^2 + 50T + 5
$79$	$ T^{20} $ T^20
$83$	$ T^{20} $ T^20
$89$	$ T^{20} + 5 T^{19} + \cdots + 1 $ T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$97$	$ T^{20} - 5 T^{17} + \cdots + 5 $ T^20 - 5T^17 - 5T^16 + 25T^14 - 75T^13 + 50T^12 - 125T^11 + 250T^10 + 125T^9 + 375T^8 - 475T^7 + 575T^6 - 620T^5 + 625T^4 - 500T^3 + 225T^2 - 50T + 5
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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:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2000.bp (of order \(50\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{50}^{9} q^{5} + \zeta_{50}^{7} q^{9} + ( - \zeta_{50}^{11} + \zeta_{50}^{3}) q^{13} + ( - \zeta_{50}^{17} - \zeta_{50}^{6}) q^{17} + \zeta_{50}^{18} q^{25} + (\zeta_{50}^{8} + \zeta_{50}^{4}) q^{29}+ \cdots + ( - \zeta_{50}^{23} - \zeta_{50}^{14}) q^{97}+O(q^{100}) \) q + z^9 * q^5 + z^7 * q^9 + (-z^11 + z^3) * q^13 + (-z^17 - z^6) * q^17 + z^18 * q^25 + (z^8 + z^4) * q^29 + (z^24 + z^5) * q^37 + (z^23 + z^21) * q^41 + z^16 * q^45 - z^10 * q^49 + (z^15 + z^2) * q^53 + (z^19 - z^12) * q^61 + (-z^20 + z^12) * q^65 + (-z^22 - z^19) * q^73 + z^14 * q^81 + (-z^15 + z) * q^85 + (-z^21 + z^20) * q^89 + (-z^23 - z^14) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{37} + 5 q^{49} + 5 q^{53} + 5 q^{65} - 5 q^{85} - 5 q^{89}+O(q^{100}) \) 20 * q + 5 * q^37 + 5 * q^49 + 5 * q^53 + 5 * q^65 - 5 * q^85 - 5 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	2000.1.bp.a

Level	$2000$
Weight	$1$
Character orbit	2000.bp
Analytic conductor	$0.998$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{50}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.998130025266\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{50})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) x^20 - x^15 + x^10 - x^5 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{50}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{50} - \cdots)\)

\(n\)	\(501\)	\(751\)	\(1377\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\zeta_{50}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 79.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
125.h	even	50	1	inner
500.n	odd	50	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( T^{20} \) T^20
$5$	\( T^{20} - T^{15} + \cdots + 1 \) T^20 - T^15 + T^10 - T^5 + 1
$7$	\( T^{20} \) T^20
$11$	\( T^{20} \) T^20
$13$	\( T^{20} + 20 T^{17} + \cdots + 5 \) T^20 + 20T^17 - 5T^16 + 150T^14 - 75T^13 + 25T^12 + 525T^11 - 375T^10 + 250T^9 + 750T^8 - 750T^7 + 650T^6 + 5T^5 + 125T^3 + 25T^2 - 25*T + 5
$17$	\( T^{20} - 5 T^{17} + \cdots + 5 \) T^20 - 5T^17 - 5T^16 + 25T^14 - 75T^13 + 50T^12 - 125T^11 + 250T^10 + 125T^9 + 375T^8 - 475T^7 + 575T^6 - 620T^5 + 625T^4 - 500T^3 + 225T^2 - 50T + 5
$19$	\( T^{20} \) T^20
$23$	\( T^{20} \) T^20
$29$	\( T^{20} + 5 T^{17} + \cdots + 1 \) T^20 + 5T^17 + 5T^16 + 2T^15 + 25T^14 + 50T^13 + 45T^12 + 145T^11 + 379T^10 + 525T^9 + 1050T^8 + 2685T^7 + 4160T^6 + 3628T^5 + 1850T^4 + 575T^3 + 110T^2 + 10*T + 1
$31$	\( T^{20} \) T^20
$37$	\( T^{20} - 5 T^{19} + \cdots + 5 \) T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5
$41$	\( T^{20} - 5 T^{17} + \cdots + 1 \) T^20 - 5T^17 + 5T^16 - 2T^15 + 25T^14 - 50T^13 + 45T^12 - 145T^11 + 379T^10 - 525T^9 + 1050T^8 - 2685T^7 + 4160T^6 - 3628T^5 + 1850T^4 - 575T^3 + 110T^2 - 10*T + 1
$43$	\( T^{20} \) T^20
$47$	\( T^{20} \) T^20
$53$	\( T^{20} - 5 T^{19} + \cdots + 5 \) T^20 - 5T^19 + 15T^18 - 35T^17 + 70T^16 - 120T^15 + 200T^14 - 325T^13 + 475T^12 - 525T^11 + 135T^10 + 525T^9 - 925T^8 + 650T^7 + 75T^6 - 240T^5 + 400T^4 - 375T^3 + 200T^2 - 50*T + 5
$59$	\( T^{20} \) T^20
$61$	\( T^{20} + 20 T^{17} + \cdots + 1 \) T^20 + 20T^17 + 5T^16 - 2T^15 + 150T^14 + 75T^13 - 5T^12 + 505T^11 + 379T^10 + 100T^9 + 800T^8 + 640T^7 + 360T^6 + 747T^5 + 600T^4 + 175T^3 + 85T^2 + 15*T + 1
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} - 5 T^{17} + \cdots + 5 \) T^20 - 5T^17 - 5T^16 + 25T^14 + 50T^13 + 25T^12 - 125T^11 - 375T^10 - 375T^9 + 500T^8 + 2525T^7 + 4400T^6 + 4380T^5 + 2750T^4 + 1125T^3 + 300T^2 + 50T + 5
$79$	\( T^{20} \) T^20
$83$	\( T^{20} \) T^20
$89$	\( T^{20} + 5 T^{19} + \cdots + 1 \) T^20 + 5T^19 + 15T^18 + 35T^17 + 70T^16 + 122T^15 + 170T^14 + 185T^13 + 165T^12 + 205T^11 + 629T^10 + 1235T^9 + 1655T^8 + 1520T^7 + 665T^6 - 252T^5 - 340T^4 - 45T^3 + 70T^2 - 10*T + 1
$97$	\( T^{20} - 5 T^{17} + \cdots + 5 \) T^20 - 5T^17 - 5T^16 + 25T^14 - 75T^13 + 50T^12 - 125T^11 + 250T^10 + 125T^9 + 375T^8 - 475T^7 + 575T^6 - 620T^5 + 625T^4 - 500T^3 + 225T^2 - 50T + 5
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