LMFDB - Newform orbit 820.1.by.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [820,1,Mod(47,820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(820, base_ring=CyclotomicField(40))

chi = DirichletCharacter(H, H._module([20, 10, 1]))

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

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

chi := DirichletCharacter("820.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 820 = 2^{2} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	820.by (of order $40$, degree $16$, 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:	$0.409233310359$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} - \cdots)$

$q$-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_{40}^{6} q^{2} + \zeta_{40}^{12} q^{4} + \zeta_{40} q^{5} - \zeta_{40}^{18} q^{8} - \zeta_{40}^{15} q^{9} - \zeta_{40}^{7} q^{10} + (\zeta_{40}^{19} + \zeta_{40}^{8}) q^{13} - \zeta_{40}^{4} q^{16} + \cdots + \zeta_{40}^{9} q^{98} +O(q^{100}) $ q - z^6 * q^2 + z^12 * q^4 + z * q^5 - z^18 * q^8 - z^15 * q^9 - z^7 * q^10 + (z^19 + z^8) * q^13 - z^4 * q^16 + (z^17 + z^4) * q^17 - z * q^18 + z^13 * q^20 + z^2 * q^25 + (-z^14 + z^5) * q^26 + (-z^17 - z^12) * q^29 + z^10 * q^32 + (-z^10 + z^3) * q^34 + z^7 * q^36 + (-z^9 - z^5) * q^37 - z^19 * q^40 + z^6 * q^41 - z^16 * q^45 - z^3 * q^49 - z^8 * q^50 + (-z^11 - 1) * q^52 + (z^11 - z^8) * q^53 + (z^18 - z^3) * q^58 + (z^16 - z^2) * q^61 - z^16 * q^64 + (z^9 - 1) * q^65 + (z^16 - z^9) * q^68 - z^13 * q^72 + (-z^13 + z^7) * q^73 + (z^15 + z^11) * q^74 - z^5 * q^80 - z^10 * q^81 - z^12 * q^82 + (z^18 + z^5) * q^85 + (-z^19 + z^14) * q^89 - z^2 * q^90 + (z^15 - z^14) * q^97 + z^9 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{4} - 4 q^{13} - 4 q^{16} + 4 q^{17} - 4 q^{29} + 4 q^{45} + 4 q^{50} - 16 q^{52} + 4 q^{53} - 4 q^{61} + 4 q^{64} - 16 q^{65} - 4 q^{68} - 4 q^{82}+O(q^{100}) $ 16 * q + 4 * q^4 - 4 * q^13 - 4 * q^16 + 4 * q^17 - 4 * q^29 + 4 * q^45 + 4 * q^50 - 16 * q^52 + 4 * q^53 - 4 * q^61 + 4 * q^64 - 16 * q^65 - 4 * q^68 - 4 * q^82

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/820\mathbb{Z}\right)^\times$.

$n$	$411$	$621$	$657$
$\chi(n)$	$-1$	$\zeta_{40}^{7}$	$-\zeta_{40}^{10}$

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} $

47.1

−0.891007	−	0.453990i
0.156434	−	0.987688i
−0.987688	+	0.156434i
−0.453990	−	0.891007i
0.453990	+	0.891007i
0.891007	−	0.453990i
0.453990	−	0.891007i
0.987688	−	0.156434i
−0.987688	−	0.156434i
−0.156434	−	0.987688i
−0.156434	+	0.987688i
0.891007	+	0.453990i
0.156434	+	0.987688i
0.987688	+	0.156434i
−0.453990	+	0.891007i
−0.891007	+	0.453990i

0.951057

−

0.309017i

0.809017

−

0.587785i

−0.891007

−

0.453990i

0.587785

−

0.809017i

0.707107

0.707107i

−0.987688

−

0.156434i

67.1

0.587785

0.809017i

−0.309017

0.951057i

0.156434

−

0.987688i

−0.951057

0.309017i

0.707107

0.707107i

0.891007

−

0.453990i

147.1

−0.587785

0.809017i

−0.309017

−

0.951057i

−0.987688

0.156434i

0.951057

0.309017i

−0.707107

−

0.707107i

0.453990

−

0.891007i

227.1

−0.951057

−

0.309017i

0.809017

0.587785i

−0.453990

−

0.891007i

−0.587785

−

0.809017i

−0.707107

−

0.707107i

0.156434

0.987688i

347.1

−0.951057

−

0.309017i

0.809017

0.587785i

0.453990

0.891007i

−0.587785

−

0.809017i

0.707107

0.707107i

−0.156434

−

0.987688i

403.1

0.951057

0.309017i

0.809017

0.587785i

0.891007

−

0.453990i

0.587785

0.809017i

−0.707107

0.707107i

0.987688

−

0.156434i

423.1

−0.951057

0.309017i

0.809017

−

0.587785i

0.453990

−

0.891007i

−0.587785

0.809017i

0.707107

−

0.707107i

−0.156434

0.987688i

427.1

−0.587785

0.809017i

−0.309017

−

0.951057i

0.987688

−

0.156434i

0.951057

0.309017i

0.707107

0.707107i

−0.453990

0.891007i

463.1

−0.587785

−

0.809017i

−0.309017

0.951057i

−0.987688

−

0.156434i

0.951057

−

0.309017i

−0.707107

0.707107i

0.453990

0.891007i

503.1

0.587785

−

0.809017i

−0.309017

−

0.951057i

−0.156434

−

0.987688i

−0.951057

−

0.309017i

−0.707107

0.707107i

−0.891007

−

0.453990i

507.1

0.587785

0.809017i

−0.309017

0.951057i

−0.156434

0.987688i

−0.951057

0.309017i

−0.707107

−

0.707107i

−0.891007

0.453990i

527.1

0.951057

−

0.309017i

0.809017

−

0.587785i

0.891007

0.453990i

0.587785

−

0.809017i

−0.707107

−

0.707107i

0.987688

0.156434i

563.1

0.587785

−

0.809017i

−0.309017

−

0.951057i

0.156434

0.987688i

−0.951057

−

0.309017i

0.707107

−

0.707107i

0.891007

0.453990i

603.1

−0.587785

−

0.809017i

−0.309017

0.951057i

0.987688

0.156434i

0.951057

−

0.309017i

0.707107

−

0.707107i

−0.453990

−

0.891007i

643.1

−0.951057

0.309017i

0.809017

−

0.587785i

−0.453990

0.891007i

−0.587785

0.809017i

−0.707107

0.707107i

0.156434

−

0.987688i

663.1

0.951057

0.309017i

0.809017

0.587785i

−0.891007

0.453990i

0.587785

0.809017i

0.707107

−

0.707107i

−0.987688

0.156434i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
205.bb	even	40	1	inner
820.by	odd	40	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	820.1.by.a	✓	16
4.b	odd	2	1	CM	820.1.by.a	✓	16
5.c	odd	4	1		820.1.bz.a	yes	16
20.e	even	4	1		820.1.bz.a	yes	16
41.h	odd	40	1		820.1.bz.a	yes	16
164.o	even	40	1		820.1.bz.a	yes	16
205.bb	even	40	1	inner	820.1.by.a	✓	16
820.by	odd	40	1	inner	820.1.by.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
820.1.by.a	✓	16	1.a	even	1	1	trivial
820.1.by.a	✓	16	4.b	odd	2	1	CM
820.1.by.a	✓	16	205.bb	even	40	1	inner
820.1.by.a	✓	16	820.by	odd	40	1	inner
820.1.bz.a	yes	16	5.c	odd	4	1
820.1.bz.a	yes	16	20.e	even	4	1
820.1.bz.a	yes	16	41.h	odd	40	1
820.1.bz.a	yes	16	164.o	even	40	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(820, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} $ (T^8 - T^6 + T^4 - T^2 + 1)^2
$3$	$ T^{16} $ T^16
$5$	$ T^{16} - T^{12} + \cdots + 1 $ T^16 - T^12 + T^8 - T^4 + 1
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} + 4 T^{15} + \cdots + 1 $ T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 44T^11 + 32T^10 + 60T^9 + 156T^8 + 284T^7 + 382T^6 + 328T^5 + 269T^4 + 68T^3 - 4T^2 - 8*T + 1
$17$	$ T^{16} - 4 T^{15} + \cdots + 1 $ T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 32T^10 - 60T^9 + 156T^8 - 284T^7 + 382T^6 - 328T^5 + 269T^4 - 68T^3 - 4T^2 + 8*T + 1
$19$	$ T^{16} $ T^16
$23$	$ T^{16} $ T^16
$29$	$ T^{16} + 4 T^{15} + \cdots + 16 $ T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 64T^11 + 112T^10 + 160T^9 + 156T^8 - 16T^7 - 128T^6 - 112T^5 - 56T^4 - 32T^3 + 16T^2 + 32*T + 16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} + 5 T^{12} + \cdots + 625 $ T^16 + 5T^12 + 150T^8 - 500*T^4 + 625
$41$	$ (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} $ (T^8 - T^6 + T^4 - T^2 + 1)^2
$43$	$ T^{16} $ T^16
$47$	$ T^{16} $ T^16
$53$	$ T^{16} - 4 T^{15} + \cdots + 1 $ T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 82T^10 - 160T^9 + 256T^8 - 304T^7 + 162T^6 + 72T^5 - 131T^4 + 32T^3 + 46T^2 + 8*T + 1
$59$	$ T^{16} $ T^16
$61$	$ (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} $ (T^8 + 2T^7 + 2T^6 + 10T^5 + 16T^4 + 10T^3 + 13T^2 + 6*T + 1)^2
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} $ (T^8 - 8T^6 + 19T^4 - 12*T^2 + 1)^2
$79$	$ T^{16} $ T^16
$83$	$ T^{16} $ T^16
$89$	$ T^{16} - 2 T^{14} + \cdots + 16 $ T^16 - 2T^14 - 4T^13 + 2T^12 - 24T^11 + 16T^10 + 48T^9 + 60T^8 - 112T^7 + 128T^6 - 112T^5 + 72T^4 - 64T^3 + 48T^2 - 32T + 16
$97$	$ T^{16} - 2 T^{14} + \cdots + 1 $ T^16 - 2T^14 + 4T^13 + 7T^12 - 16T^11 + 6T^10 - 8T^9 - 80T^8 + 72T^7 + 118T^6 + 92T^5 + 222T^4 + 144T^3 + 68T^2 + 12T + 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:	\( N \)	\(=\)	\( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	820.by (of order \(40\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{40}^{6} q^{2} + \zeta_{40}^{12} q^{4} + \zeta_{40} q^{5} - \zeta_{40}^{18} q^{8} - \zeta_{40}^{15} q^{9} - \zeta_{40}^{7} q^{10} + (\zeta_{40}^{19} + \zeta_{40}^{8}) q^{13} - \zeta_{40}^{4} q^{16} + \cdots + \zeta_{40}^{9} q^{98} +O(q^{100}) \) q - z^6 * q^2 + z^12 * q^4 + z * q^5 - z^18 * q^8 - z^15 * q^9 - z^7 * q^10 + (z^19 + z^8) * q^13 - z^4 * q^16 + (z^17 + z^4) * q^17 - z * q^18 + z^13 * q^20 + z^2 * q^25 + (-z^14 + z^5) * q^26 + (-z^17 - z^12) * q^29 + z^10 * q^32 + (-z^10 + z^3) * q^34 + z^7 * q^36 + (-z^9 - z^5) * q^37 - z^19 * q^40 + z^6 * q^41 - z^16 * q^45 - z^3 * q^49 - z^8 * q^50 + (-z^11 - 1) * q^52 + (z^11 - z^8) * q^53 + (z^18 - z^3) * q^58 + (z^16 - z^2) * q^61 - z^16 * q^64 + (z^9 - 1) * q^65 + (z^16 - z^9) * q^68 - z^13 * q^72 + (-z^13 + z^7) * q^73 + (z^15 + z^11) * q^74 - z^5 * q^80 - z^10 * q^81 - z^12 * q^82 + (z^18 + z^5) * q^85 + (-z^19 + z^14) * q^89 - z^2 * q^90 + (z^15 - z^14) * q^97 + z^9 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{4} - 4 q^{13} - 4 q^{16} + 4 q^{17} - 4 q^{29} + 4 q^{45} + 4 q^{50} - 16 q^{52} + 4 q^{53} - 4 q^{61} + 4 q^{64} - 16 q^{65} - 4 q^{68} - 4 q^{82}+O(q^{100}) \) 16 * q + 4 * q^4 - 4 * q^13 - 4 * q^16 + 4 * q^17 - 4 * q^29 + 4 * q^45 + 4 * q^50 - 16 * q^52 + 4 * q^53 - 4 * q^61 + 4 * q^64 - 16 * q^65 - 4 * q^68 - 4 * q^82

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	820.1.by.a

Level	$820$
Weight	$1$
Character orbit	820.by
Analytic conductor	$0.409$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{40}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.409233310359\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{40})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{40}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

\(n\)	\(411\)	\(621\)	\(657\)
\(\chi(n)\)	\(-1\)	\(\zeta_{40}^{7}\)	\(-\zeta_{40}^{10}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 47.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
205.bb	even	40	1	inner
820.by	odd	40	1	inner

$p$	$F_p(T)$
$2$	\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) (T^8 - T^6 + T^4 - T^2 + 1)^2
$3$	\( T^{16} \) T^16
$5$	\( T^{16} - T^{12} + \cdots + 1 \) T^16 - T^12 + T^8 - T^4 + 1
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} + 4 T^{15} + \cdots + 1 \) T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 44T^11 + 32T^10 + 60T^9 + 156T^8 + 284T^7 + 382T^6 + 328T^5 + 269T^4 + 68T^3 - 4T^2 - 8*T + 1
$17$	\( T^{16} - 4 T^{15} + \cdots + 1 \) T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 32T^10 - 60T^9 + 156T^8 - 284T^7 + 382T^6 - 328T^5 + 269T^4 - 68T^3 - 4T^2 + 8*T + 1
$19$	\( T^{16} \) T^16
$23$	\( T^{16} \) T^16
$29$	\( T^{16} + 4 T^{15} + \cdots + 16 \) T^16 + 4T^15 + 10T^14 + 20T^13 + 34T^12 + 64T^11 + 112T^10 + 160T^9 + 156T^8 - 16T^7 - 128T^6 - 112T^5 - 56T^4 - 32T^3 + 16T^2 + 32*T + 16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} + 5 T^{12} + \cdots + 625 \) T^16 + 5T^12 + 150T^8 - 500*T^4 + 625
$41$	\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) (T^8 - T^6 + T^4 - T^2 + 1)^2
$43$	\( T^{16} \) T^16
$47$	\( T^{16} \) T^16
$53$	\( T^{16} - 4 T^{15} + \cdots + 1 \) T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 82T^10 - 160T^9 + 256T^8 - 304T^7 + 162T^6 + 72T^5 - 131T^4 + 32T^3 + 46T^2 + 8*T + 1
$59$	\( T^{16} \) T^16
$61$	\( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) (T^8 + 2T^7 + 2T^6 + 10T^5 + 16T^4 + 10T^3 + 13T^2 + 6*T + 1)^2
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) (T^8 - 8T^6 + 19T^4 - 12*T^2 + 1)^2
$79$	\( T^{16} \) T^16
$83$	\( T^{16} \) T^16
$89$	\( T^{16} - 2 T^{14} + \cdots + 16 \) T^16 - 2T^14 - 4T^13 + 2T^12 - 24T^11 + 16T^10 + 48T^9 + 60T^8 - 112T^7 + 128T^6 - 112T^5 + 72T^4 - 64T^3 + 48T^2 - 32T + 16
$97$	\( T^{16} - 2 T^{14} + \cdots + 1 \) T^16 - 2T^14 + 4T^13 + 7T^12 - 16T^11 + 6T^10 - 8T^9 - 80T^8 + 72T^7 + 118T^6 + 92T^5 + 222T^4 + 144T^3 + 68T^2 + 12T + 1
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