LMFDB - Newform orbit 1320.1.dk.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1320,1,Mod(173,1320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1320, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 10, 10, 15, 6]))

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

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

chi := DirichletCharacter("1320.173");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1320.dk (of order $20$ , degree $8$ , 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:	$0.658765816676$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

$q$ -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.

$f(q)$	$=$	$q - \zeta_{40}^{17} q^{2} + \zeta_{40} q^{3} - \zeta_{40}^{14} q^{4} + \zeta_{40}^{19} q^{5} - \zeta_{40}^{18} q^{6} + ( - \zeta_{40}^{12} - \zeta_{40}^{6}) q^{7} - \zeta_{40}^{11} q^{8} + \zeta_{40}^{2} q^{9} + \cdots + \zeta_{40}^{17} q^{99} +O(q^{100})$ q - z^17 * q^2 + z * q^3 - z^14 * q^4 + z^19 * q^5 - z^18 * q^6 + (-z^12 - z^6) * q^7 - z^11 * q^8 + z^2 * q^9 + z^16 * q^10 + z^15 * q^11 - z^15 * q^12 + (-z^9 - z^3) * q^14 - q^15 - z^8 * q^16 - z^19 * q^18 + z^13 * q^20 + (-z^13 - z^7) * q^21 + z^12 * q^22 - z^12 * q^24 - z^18 * q^25 + z^3 * q^27 + (-z^6 - 1) * q^28 + (z^5 + z^3) * q^29 + z^17 * q^30 + (z^14 + z^10) * q^31 - z^5 * q^32 + z^16 * q^33 + (z^11 + z^5) * q^35 - z^16 * q^36 + z^10 * q^40 + (-z^10 - z^4) * q^42 + z^9 * q^44 - z * q^45 - z^9 * q^48 + (z^18 + z^12 - z^4) * q^49 - z^15 * q^50 + (z^9 - z^5) * q^53 + q^54 - z^14 * q^55 + (z^17 - z^3) * q^56 + (z^2 + 1) * q^58 + (z^7 - z) * q^59 + z^14 * q^60 + (z^11 + z^7) * q^62 + (-z^14 - z^8) * q^63 - z^2 * q^64 + z^13 * q^66 + (z^8 + z^2) * q^70 - z^13 * q^72 + (-z^10 + z^8) * q^73 - z^19 * q^75 + (z^7 + z) * q^77 + (-z^16 - z^8) * q^79 + z^7 * q^80 + z^4 * q^81 + (z^19 - z^7) * q^83 + (-z^7 - z) * q^84 + (z^6 + z^4) * q^87 + z^6 * q^88 + z^18 * q^90 + (z^15 + z^11) * q^93 - z^6 * q^96 + (-z^18 + z^16) * q^97 + (z^15 + z^9 - z) * q^98 + z^17 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 4 q^{7} - 4 q^{10} - 16 q^{15} + 4 q^{16} + 4 q^{22} - 4 q^{24} - 16 q^{28} - 4 q^{33} + 4 q^{36} - 4 q^{42} + 16 q^{54} + 16 q^{58} + 4 q^{63} - 4 q^{70} - 4 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87}+ \cdots - 4 q^{97}+O(q^{100})$ 16 * q - 4 * q^7 - 4 * q^10 - 16 * q^15 + 4 * q^16 + 4 * q^22 - 4 * q^24 - 16 * q^28 - 4 * q^33 + 4 * q^36 - 4 * q^42 + 16 * q^54 + 16 * q^58 + 4 * q^63 - 4 * q^70 - 4 * q^73 + 8 * q^79 + 4 * q^81 + 4 * q^87 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times$ .

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$-1$	$-1$	$1$	$-\zeta_{40}^{10}$	$\zeta_{40}^{4}$

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}

173.1

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

−0.987688

−

0.156434i

0.453990

−

0.891007i

0.951057

0.309017i

−0.453990

−

0.891007i

−0.587785

0.809017i

−1.76007

0.896802i

−0.891007

−

0.453990i

−0.587785

−

0.809017i

0.309017

0.951057i

173.2

0.987688

0.156434i

−0.453990

0.891007i

0.951057

0.309017i

0.453990

0.891007i

−0.587785

0.809017i

−1.76007

0.896802i

0.891007

0.453990i

−0.587785

−

0.809017i

0.309017

0.951057i

293.1

−0.156434

−

0.987688i

−0.891007

0.453990i

−0.951057

0.309017i

0.891007

0.453990i

0.587785

0.809017i

0.142040

−

0.278768i

0.453990

0.891007i

0.587785

−

0.809017i

0.309017

−

0.951057i

293.2

0.156434

0.987688i

0.891007

−

0.453990i

−0.951057

0.309017i

−0.891007

−

0.453990i

0.587785

0.809017i

0.142040

−

0.278768i

−0.453990

−

0.891007i

0.587785

−

0.809017i

0.309017

−

0.951057i

413.1

−0.453990

0.891007i

0.156434

0.987688i

−0.587785

−

0.809017i

−0.156434

0.987688i

−0.951057

−

0.309017i

0.896802

0.142040i

0.987688

−

0.156434i

−0.951057

0.309017i

−0.809017

−

0.587785i

413.2

0.453990

−

0.891007i

−0.156434

−

0.987688i

−0.587785

−

0.809017i

0.156434

−

0.987688i

−0.951057

−

0.309017i

0.896802

0.142040i

−0.987688

0.156434i

−0.951057

0.309017i

−0.809017

−

0.587785i

437.1

−0.156434

0.987688i

−0.891007

−

0.453990i

−0.951057

−

0.309017i

0.891007

−

0.453990i

0.587785

−

0.809017i

0.142040

0.278768i

0.453990

−

0.891007i

0.587785

0.809017i

0.309017

0.951057i

437.2

0.156434

−

0.987688i

0.891007

0.453990i

−0.951057

−

0.309017i

−0.891007

0.453990i

0.587785

−

0.809017i

0.142040

0.278768i

−0.453990

0.891007i

0.587785

0.809017i

0.309017

0.951057i

557.1

−0.987688

0.156434i

0.453990

0.891007i

0.951057

−

0.309017i

−0.453990

0.891007i

−0.587785

−

0.809017i

−1.76007

−

0.896802i

−0.891007

0.453990i

−0.587785

0.809017i

0.309017

−

0.951057i

557.2

0.987688

−

0.156434i

−0.453990

−

0.891007i

0.951057

−

0.309017i

0.453990

−

0.891007i

−0.587785

−

0.809017i

−1.76007

−

0.896802i

0.891007

−

0.453990i

−0.587785

0.809017i

0.309017

−

0.951057i

677.1

−0.891007

−

0.453990i

−0.987688

0.156434i

0.587785

0.809017i

0.987688

0.156434i

0.951057

0.309017i

−0.278768

1.76007i

−0.156434

−

0.987688i

0.951057

−

0.309017i

−0.809017

−

0.587785i

677.2

0.891007

0.453990i

0.987688

−

0.156434i

0.587785

0.809017i

−0.987688

−

0.156434i

0.951057

0.309017i

−0.278768

1.76007i

0.156434

0.987688i

0.951057

−

0.309017i

−0.809017

−

0.587785i

893.1

−0.891007

0.453990i

−0.987688

−

0.156434i

0.587785

−

0.809017i

0.987688

−

0.156434i

0.951057

−

0.309017i

−0.278768

−

1.76007i

−0.156434

0.987688i

0.951057

0.309017i

−0.809017

0.587785i

893.2

0.891007

−

0.453990i

0.987688

0.156434i

0.587785

−

0.809017i

−0.987688

0.156434i

0.951057

−

0.309017i

−0.278768

−

1.76007i

0.156434

−

0.987688i

0.951057

0.309017i

−0.809017

0.587785i

1157.1

−0.453990

−

0.891007i

0.156434

−

0.987688i

−0.587785

0.809017i

−0.156434

−

0.987688i

−0.951057

0.309017i

0.896802

−

0.142040i

0.987688

0.156434i

−0.951057

−

0.309017i

−0.809017

0.587785i

1157.2

0.453990

0.891007i

−0.156434

0.987688i

−0.587785

0.809017i

0.156434

0.987688i

−0.951057

0.309017i

0.896802

−

0.142040i

−0.987688

−

0.156434i

−0.951057

−

0.309017i

−0.809017

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6})$
3.b	odd	2	1	inner
8.b	even	2	1	inner
55.l	even	20	1	inner
165.u	odd	20	1	inner
440.bu	even	20	1	inner
1320.dk	odd	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1320.1.dk.a	✓	16
3.b	odd	2	1	inner	1320.1.dk.a	✓	16
5.c	odd	4	1		1320.1.dk.b	yes	16
8.b	even	2	1	inner	1320.1.dk.a	✓	16
11.d	odd	10	1		1320.1.dk.b	yes	16
15.e	even	4	1		1320.1.dk.b	yes	16
24.h	odd	2	1	CM	1320.1.dk.a	✓	16
33.f	even	10	1		1320.1.dk.b	yes	16
40.i	odd	4	1		1320.1.dk.b	yes	16
55.l	even	20	1	inner	1320.1.dk.a	✓	16
88.p	odd	10	1		1320.1.dk.b	yes	16
120.w	even	4	1		1320.1.dk.b	yes	16
165.u	odd	20	1	inner	1320.1.dk.a	✓	16
264.u	even	10	1		1320.1.dk.b	yes	16
440.bu	even	20	1	inner	1320.1.dk.a	✓	16
1320.dk	odd	20	1	inner	1320.1.dk.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1320.1.dk.a	✓	16	1.a	even	1	1	trivial
1320.1.dk.a	✓	16	3.b	odd	2	1	inner
1320.1.dk.a	✓	16	8.b	even	2	1	inner
1320.1.dk.a	✓	16	24.h	odd	2	1	CM
1320.1.dk.a	✓	16	55.l	even	20	1	inner
1320.1.dk.a	✓	16	165.u	odd	20	1	inner
1320.1.dk.a	✓	16	440.bu	even	20	1	inner
1320.1.dk.a	✓	16	1320.dk	odd	20	1	inner
1320.1.dk.b	yes	16	5.c	odd	4	1
1320.1.dk.b	yes	16	11.d	odd	10	1
1320.1.dk.b	yes	16	15.e	even	4	1
1320.1.dk.b	yes	16	33.f	even	10	1
1320.1.dk.b	yes	16	40.i	odd	4	1
1320.1.dk.b	yes	16	88.p	odd	10	1
1320.1.dk.b	yes	16	120.w	even	4	1
1320.1.dk.b	yes	16	264.u	even	10	1

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{8} + 2T_{7}^{7} + 2T_{7}^{6} - 4T_{7}^{4} - 10T_{7}^{3} + 13T_{7}^{2} - 4T_{7} + 1$ T7^8 + 2*T7^7 + 2*T7^6 - 4*T7^4 - 10*T7^3 + 13*T7^2 - 4*T7 + 1 acting on $S_{1}^{\mathrm{new}}(1320, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16} - T^{12} + \cdots + 1$ T^16 - T^12 + T^8 - T^4 + 1
$3$	$T^{16} - T^{12} + \cdots + 1$ T^16 - T^12 + T^8 - T^4 + 1
$5$	$T^{16} - T^{12} + \cdots + 1$ T^16 - T^12 + T^8 - T^4 + 1
$7$	$(T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2}$ (T^8 + 2T^7 + 2T^6 - 4T^4 - 10T^3 + 13T^2 - 4T + 1)^2
$11$	$(T^{4} + 1)^{4}$ (T^4 + 1)^4
$13$	$T^{16}$ T^16
$17$	$T^{16}$ T^16
$19$	$T^{16}$ T^16
$23$	$T^{16}$ T^16
$29$	$T^{16} + 4 T^{14} + \cdots + 1$ T^16 + 4T^14 + 17T^12 + 72T^10 + 230T^8 + 228T^6 + 92T^4 - 4*T^2 + 1
$31$	$(T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2}$ (T^8 + 5T^6 + 10T^4 + 25)^2
$37$	$T^{16}$ T^16
$41$	$T^{16}$ T^16
$43$	$T^{16}$ T^16
$47$	$T^{16}$ T^16
$53$	$T^{16} - 11 T^{12} + \cdots + 1$ T^16 - 11T^12 + 46T^8 + 4*T^4 + 1
$59$	$T^{16} - 4 T^{14} + \cdots + 1$ T^16 - 4T^14 + 12T^12 - 32T^10 + 150T^8 + 32T^6 + 97T^4 - 16*T^2 + 1
$61$	$T^{16}$ T^16
$67$	$T^{16}$ T^16
$71$	$T^{16}$ T^16
$73$	$(T^{8} + 2 T^{7} + 7 T^{6} + \cdots + 1)^{2}$ (T^8 + 2T^7 + 7T^6 + 10T^5 + 16T^4 + 10T^3 - 2T^2 - 4*T + 1)^2
$79$	$(T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{4}$ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^4
$83$	$T^{16} + 4 T^{12} + \cdots + 1$ T^16 + 4T^12 + 46T^8 - 11*T^4 + 1
$89$	$T^{16}$ T^16
$97$	$(T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2}$ (T^8 + 2T^7 + 2T^6 - 4T^4 - 10T^3 + 13T^2 - 4T + 1)^2
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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 173.2
Significant digits:
Format:

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	1320.1.dk.a

Level	$1320$
Weight	$1$
Character orbit	1320.dk
Analytic conductor	$0.659$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{20}$
CM discriminant	-24
Inner twists	$8$