LMFDB - Newform orbit 2169.1.de.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,1,Mod(28,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, base_ring=CyclotomicField(80))

chi = DirichletCharacter(H, H._module([0, 47]))

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

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

chi := DirichletCharacter("2169.28");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2169.de (of order $80$, degree $32$, 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:	$1.08247201240$
Analytic rank:	$0$
Dimension:	$32$
Coefficient field:	$\Q(\zeta_{80})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} - x^{24} + x^{16} - x^{8} + 1 $ x^32 - x^24 + x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{80}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{80} - \cdots)$

$q$-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_{80}^{30} q^{4} + ( - \zeta_{80}^{19} + \zeta_{80}^{18}) q^{7}+O(q^{10}) $ q + z^30 * q^4 + (-z^19 + z^18) * q^7	$ q + \zeta_{80}^{30} q^{4} + ( - \zeta_{80}^{19} + \zeta_{80}^{18}) q^{7} + ( - \zeta_{80}^{34} + \zeta_{80}^{25}) q^{13} - \zeta_{80}^{20} q^{16} + ( - \zeta_{80}^{23} + \zeta_{80}^{2}) q^{19} + \zeta_{80}^{26} q^{25} + (\zeta_{80}^{9} - \zeta_{80}^{8}) q^{28} + (\zeta_{80}^{16} + \zeta_{80}^{11}) q^{31} + ( - \zeta_{80}^{39} + \zeta_{80}^{22}) q^{37} + (\zeta_{80}^{35} + \zeta_{80}^{28}) q^{43} + (\zeta_{80}^{38} + \cdots + \zeta_{80}^{36}) q^{49} + \cdots + (\zeta_{80}^{21} + \zeta_{80}^{7}) q^{97} +O(q^{100}) $ q + z^30 * q^4 + (-z^19 + z^18) * q^7 + (-z^34 + z^25) * q^13 - z^20 * q^16 + (-z^23 + z^2) * q^19 + z^26 * q^25 + (z^9 - z^8) * q^28 + (z^16 + z^11) * q^31 + (-z^39 + z^22) * q^37 + (z^35 + z^28) * q^43 + (z^38 - z^37 + z^36) * q^49 + (z^24 - z^15) * q^52 + (z^31 + z^3) * q^61 + z^10 * q^64 + (-z^33 - z^29) * q^67 + (z^37 - z^24) * q^73 + (z^32 + z^13) * q^76 + (-z^31 + z^15) * q^79 + (-z^13 + z^12 + z^4 - z^3) * q^91 + (z^21 + z^7) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 8 q^{28} - 8 q^{31} + 8 q^{52} - 8 q^{73} - 8 q^{76}+O(q^{100}) $ 32 * q - 8 * q^28 - 8 * q^31 + 8 * q^52 - 8 * q^73 - 8 * q^76

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2169\mathbb{Z}\right)^\times$.

$n$	$730$	$965$
$\chi(n)$	$-\zeta_{80}^{37}$	$1$

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} $

28.1

0.649448	+	0.760406i
0.233445	−	0.972370i
−0.760406	+	0.649448i
0.233445	+	0.972370i
−0.852640	+	0.522499i
0.972370	−	0.233445i
−0.972370	−	0.233445i
−0.522499	−	0.852640i
−0.0784591	−	0.996917i
−0.996917	+	0.0784591i
−0.852640	−	0.522499i
−0.760406	−	0.649448i
−0.649448	+	0.760406i
−0.0784591	+	0.996917i
0.0784591	−	0.996917i
0.649448	−	0.760406i
0.760406	+	0.649448i
0.852640	+	0.522499i
0.996917	−	0.0784591i
0.0784591	+	0.996917i

0.707107

0.707107i

−0.227282

0.805883i

73.1

−0.707107

−

0.707107i

−0.518379

1.12445i

136.1

−0.707107

−

0.707107i

1.63714

−

0.916840i

208.1

−0.707107

0.707107i

−0.518379

−

1.12445i

262.1

−0.707107

0.707107i

−1.41351

1.30663i

298.1

0.707107

−

0.707107i

−0.220545

−

0.0813634i

334.1

0.707107

0.707107i

−0.687436

−

1.86338i

343.1

0.707107

−

0.707107i

1.74365

0.0685081i

379.1

0.707107

0.707107i

−1.15335

0.909229i

397.1

−0.707107

−

0.707107i

0.234894

−

1.98461i

505.1

−0.707107

−

0.707107i

−1.41351

−

1.30663i

622.1

−0.707107

0.707107i

1.63714

0.916840i

766.1

0.707107

−

0.707107i

−1.74809

0.493014i

847.1

0.707107

−

0.707107i

−1.15335

−

0.909229i

1081.1

0.707107

−

0.707107i

0.840483

−

1.06615i

1162.1

0.707107

−

0.707107i

−0.227282

−

0.805883i

1306.1

−0.707107

0.707107i

0.338240

−

0.603972i

1423.1

−0.707107

−

0.707107i

−0.368508

0.398650i

1531.1

−0.707107

−

0.707107i

0.0779754

0.00922899i

1549.1

0.707107

0.707107i

0.840483

1.06615i

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
241.r	odd	80	1	inner
723.bj	even	80	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2169.1.de.a	✓	32
3.b	odd	2	1	CM	2169.1.de.a	✓	32
241.r	odd	80	1	inner	2169.1.de.a	✓	32
723.bj	even	80	1	inner	2169.1.de.a	✓	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2169.1.de.a	✓	32	1.a	even	1	1	trivial
2169.1.de.a	✓	32	3.b	odd	2	1	CM
2169.1.de.a	✓	32	241.r	odd	80	1	inner
2169.1.de.a	✓	32	723.bj	even	80	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2169, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{32} $ T^32
$3$	$ T^{32} $ T^32
$5$	$ T^{32} $ T^32
$7$	$ T^{32} - 8 T^{29} + \cdots + 1 $ T^32 - 8T^29 - 2T^28 + 52T^26 + 40T^25 + 2T^24 - 320T^23 - 464T^22 + 104T^21 + 1936T^20 - 992T^19 + 288T^18 - 3912T^17 + 7068T^16 - 6240T^15 + 9248T^14 - 21496T^13 + 20522T^12 - 10272T^11 + 43484T^10 + 18064T^9 + 35885T^8 + 17088T^7 + 14592T^6 + 7224T^5 + 3908T^4 + 736T^3 - 40T^2 - 16T + 1
$11$	$ T^{32} $ T^32
$13$	$ T^{32} - 2 T^{28} + \cdots + 1 $ T^32 - 2T^28 + 16T^27 - 20T^26 + 8T^25 + 7T^24 - 64T^23 + 336T^22 + 1008T^21 + 646T^20 - 112T^19 + 1128T^18 + 4680T^17 + 7600T^16 - 6464T^15 - 9576T^14 - 2000T^13 + 34934T^12 + 50176T^11 + 48692T^10 + 31504T^9 + 22750T^8 + 11296T^7 + 9904T^6 + 10016T^5 + 3524T^4 + 512T^3 + 196T^2 + 24T + 1
$17$	$ T^{32} $ T^32
$19$	$ T^{32} + 8 T^{29} + \cdots + 1 $ T^32 + 8T^29 - 2T^28 + 52T^26 - 40T^25 + 2T^24 + 320T^23 - 464T^22 + 96T^21 + 1936T^20 - 4288T^19 + 2148T^18 + 3512T^17 - 2997T^16 - 1440T^15 + 7528T^14 + 4416T^13 - 868T^12 + 12352T^11 + 23844T^10 + 18456T^9 + 27730T^8 + 55712T^7 + 59992T^6 + 34856T^5 + 11578T^4 + 2304T^3 + 280T^2 + 16T + 1
$23$	$ T^{32} $ T^32
$29$	$ T^{32} $ T^32
$31$	$ T^{32} + 8 T^{31} + \cdots + 16 $ T^32 + 8T^31 + 36T^30 + 120T^29 + 330T^28 + 784T^27 + 1652T^26 + 3144T^25 + 5474T^24 + 8784T^23 + 13184T^22 + 18944T^21 + 26544T^20 + 35504T^19 + 39264T^18 + 25904T^17 - 10980T^16 - 55712T^15 - 65936T^14 - 16576T^13 + 70784T^12 + 140736T^11 + 147472T^10 + 91264T^9 + 21448T^8 - 11584T^7 - 6944T^6 - 1344T^5 - 496T^4 - 320T^3 + 32T^2 + 64*T + 16
$37$	$ T^{32} - 2 T^{28} + \cdots + 1 $ T^32 - 2T^28 + 16T^27 + 80T^26 + 8T^25 + 2T^24 - 64T^23 + 136T^22 + 928T^21 + 2006T^20 + 368T^19 - 1432T^18 + 1600T^17 + 7600T^16 + 14016T^15 + 18704T^14 + 13840T^13 + 37974T^12 + 49376T^11 + 50512T^10 + 48824T^9 + 45805T^8 + 31616T^7 + 8824T^6 - 864T^5 - 1606T^4 - 448T^3 + 196T^2 - 16T + 1
$41$	$ T^{32} $ T^32
$43$	$ T^{32} - 4 T^{30} + \cdots + 1 $ T^32 - 4T^30 + 10T^28 - 8T^27 - 20T^26 + 72T^25 + 39T^24 + 208T^23 - 992T^21 - 814T^20 + 1976T^19 + 2052T^18 - 2616T^17 + 6208T^16 - 3088T^15 + 1948T^14 - 6720T^13 - 14878T^12 + 18672T^11 + 22664T^10 + 17296T^9 + 7470T^8 + 4256T^7 + 2756T^6 - 1424T^5 + 3084T^4 - 1336T^3 + 244T^2 - 16T + 1
$47$	$ T^{32} $ T^32
$53$	$ T^{32} $ T^32
$59$	$ T^{32} $ T^32
$61$	$ T^{32} + 40 T^{28} + \cdots + 1 $ T^32 + 40T^28 + 644T^24 + 5160T^20 + 19246T^16 - 16560T^12 + 11189T^8 - 200*T^4 + 1
$67$	$ T^{32} + 40 T^{28} + \cdots + 1 $ T^32 + 40T^28 + 644T^24 + 5160T^20 + 19246T^16 - 16560T^12 + 11189T^8 - 200*T^4 + 1
$71$	$ T^{32} $ T^32
$73$	$ T^{32} + 8 T^{31} + \cdots + 1 $ T^32 + 8T^31 + 36T^30 + 120T^29 + 330T^28 + 784T^27 + 1652T^26 + 3144T^25 + 5474T^24 + 8824T^23 + 13324T^22 + 18944T^21 + 24864T^20 + 27104T^19 + 19804T^18 + 2344T^17 - 10980T^16 + 9168T^15 + 72764T^14 + 149184T^13 + 182014T^12 + 135696T^11 + 49992T^10 - 10936T^9 - 26227T^8 - 23424T^7 - 14364T^6 - 3024T^5 + 3004T^4 + 1640T^3 + 312T^2 + 24*T + 1
$79$	$ T^{32} - 275 T^{24} + \cdots + 390625 $ T^32 - 275T^24 + 28750T^16 + 62500*T^8 + 390625
$83$	$ T^{32} $ T^32
$89$	$ T^{32} $ T^32
$97$	$ T^{32} - 12 T^{28} + \cdots + 1 $ T^32 - 12T^28 + 140T^24 - 1632T^20 + 40T^18 + 23574T^16 + 70240T^14 + 93152T^12 + 61360T^10 + 13665T^8 - 5280T^6 + 1872T^4 - 80T^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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2169.de (of order \(80\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{80}^{30} q^{4} + ( - \zeta_{80}^{19} + \zeta_{80}^{18}) q^{7}+O(q^{10}) \) q + z^30 * q^4 + (-z^19 + z^18) * q^7	\( q + \zeta_{80}^{30} q^{4} + ( - \zeta_{80}^{19} + \zeta_{80}^{18}) q^{7} + ( - \zeta_{80}^{34} + \zeta_{80}^{25}) q^{13} - \zeta_{80}^{20} q^{16} + ( - \zeta_{80}^{23} + \zeta_{80}^{2}) q^{19} + \zeta_{80}^{26} q^{25} + (\zeta_{80}^{9} - \zeta_{80}^{8}) q^{28} + (\zeta_{80}^{16} + \zeta_{80}^{11}) q^{31} + ( - \zeta_{80}^{39} + \zeta_{80}^{22}) q^{37} + (\zeta_{80}^{35} + \zeta_{80}^{28}) q^{43} + (\zeta_{80}^{38} + \cdots + \zeta_{80}^{36}) q^{49} + \cdots + (\zeta_{80}^{21} + \zeta_{80}^{7}) q^{97} +O(q^{100}) \) q + z^30 * q^4 + (-z^19 + z^18) * q^7 + (-z^34 + z^25) * q^13 - z^20 * q^16 + (-z^23 + z^2) * q^19 + z^26 * q^25 + (z^9 - z^8) * q^28 + (z^16 + z^11) * q^31 + (-z^39 + z^22) * q^37 + (z^35 + z^28) * q^43 + (z^38 - z^37 + z^36) * q^49 + (z^24 - z^15) * q^52 + (z^31 + z^3) * q^61 + z^10 * q^64 + (-z^33 - z^29) * q^67 + (z^37 - z^24) * q^73 + (z^32 + z^13) * q^76 + (-z^31 + z^15) * q^79 + (-z^13 + z^12 + z^4 - z^3) * q^91 + (z^21 + z^7) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 8 q^{28} - 8 q^{31} + 8 q^{52} - 8 q^{73} - 8 q^{76}+O(q^{100}) \) 32 * q - 8 * q^28 - 8 * q^31 + 8 * q^52 - 8 * q^73 - 8 * q^76

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	2169.1.de.a

Level	$2169$
Weight	$1$
Character orbit	2169.de
Analytic conductor	$1.082$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{80}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.08247201240\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Coefficient field:	\(\Q(\zeta_{80})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{32} - x^{24} + x^{16} - x^{8} + 1 \) x^32 - x^24 + x^16 - x^8 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{80}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{80} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 28.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
241.r	odd	80	1	inner
723.bj	even	80	1	inner

$p$	$F_p(T)$
$2$	\( T^{32} \) T^32
$3$	\( T^{32} \) T^32
$5$	\( T^{32} \) T^32
$7$	\( T^{32} - 8 T^{29} + \cdots + 1 \) T^32 - 8T^29 - 2T^28 + 52T^26 + 40T^25 + 2T^24 - 320T^23 - 464T^22 + 104T^21 + 1936T^20 - 992T^19 + 288T^18 - 3912T^17 + 7068T^16 - 6240T^15 + 9248T^14 - 21496T^13 + 20522T^12 - 10272T^11 + 43484T^10 + 18064T^9 + 35885T^8 + 17088T^7 + 14592T^6 + 7224T^5 + 3908T^4 + 736T^3 - 40T^2 - 16T + 1
$11$	\( T^{32} \) T^32
$13$	\( T^{32} - 2 T^{28} + \cdots + 1 \) T^32 - 2T^28 + 16T^27 - 20T^26 + 8T^25 + 7T^24 - 64T^23 + 336T^22 + 1008T^21 + 646T^20 - 112T^19 + 1128T^18 + 4680T^17 + 7600T^16 - 6464T^15 - 9576T^14 - 2000T^13 + 34934T^12 + 50176T^11 + 48692T^10 + 31504T^9 + 22750T^8 + 11296T^7 + 9904T^6 + 10016T^5 + 3524T^4 + 512T^3 + 196T^2 + 24T + 1
$17$	\( T^{32} \) T^32
$19$	\( T^{32} + 8 T^{29} + \cdots + 1 \) T^32 + 8T^29 - 2T^28 + 52T^26 - 40T^25 + 2T^24 + 320T^23 - 464T^22 + 96T^21 + 1936T^20 - 4288T^19 + 2148T^18 + 3512T^17 - 2997T^16 - 1440T^15 + 7528T^14 + 4416T^13 - 868T^12 + 12352T^11 + 23844T^10 + 18456T^9 + 27730T^8 + 55712T^7 + 59992T^6 + 34856T^5 + 11578T^4 + 2304T^3 + 280T^2 + 16T + 1
$23$	\( T^{32} \) T^32
$29$	\( T^{32} \) T^32
$31$	\( T^{32} + 8 T^{31} + \cdots + 16 \) T^32 + 8T^31 + 36T^30 + 120T^29 + 330T^28 + 784T^27 + 1652T^26 + 3144T^25 + 5474T^24 + 8784T^23 + 13184T^22 + 18944T^21 + 26544T^20 + 35504T^19 + 39264T^18 + 25904T^17 - 10980T^16 - 55712T^15 - 65936T^14 - 16576T^13 + 70784T^12 + 140736T^11 + 147472T^10 + 91264T^9 + 21448T^8 - 11584T^7 - 6944T^6 - 1344T^5 - 496T^4 - 320T^3 + 32T^2 + 64*T + 16
$37$	\( T^{32} - 2 T^{28} + \cdots + 1 \) T^32 - 2T^28 + 16T^27 + 80T^26 + 8T^25 + 2T^24 - 64T^23 + 136T^22 + 928T^21 + 2006T^20 + 368T^19 - 1432T^18 + 1600T^17 + 7600T^16 + 14016T^15 + 18704T^14 + 13840T^13 + 37974T^12 + 49376T^11 + 50512T^10 + 48824T^9 + 45805T^8 + 31616T^7 + 8824T^6 - 864T^5 - 1606T^4 - 448T^3 + 196T^2 - 16T + 1
$41$	\( T^{32} \) T^32
$43$	\( T^{32} - 4 T^{30} + \cdots + 1 \) T^32 - 4T^30 + 10T^28 - 8T^27 - 20T^26 + 72T^25 + 39T^24 + 208T^23 - 992T^21 - 814T^20 + 1976T^19 + 2052T^18 - 2616T^17 + 6208T^16 - 3088T^15 + 1948T^14 - 6720T^13 - 14878T^12 + 18672T^11 + 22664T^10 + 17296T^9 + 7470T^8 + 4256T^7 + 2756T^6 - 1424T^5 + 3084T^4 - 1336T^3 + 244T^2 - 16T + 1
$47$	\( T^{32} \) T^32
$53$	\( T^{32} \) T^32
$59$	\( T^{32} \) T^32
$61$	\( T^{32} + 40 T^{28} + \cdots + 1 \) T^32 + 40T^28 + 644T^24 + 5160T^20 + 19246T^16 - 16560T^12 + 11189T^8 - 200*T^4 + 1
$67$	\( T^{32} + 40 T^{28} + \cdots + 1 \) T^32 + 40T^28 + 644T^24 + 5160T^20 + 19246T^16 - 16560T^12 + 11189T^8 - 200*T^4 + 1
$71$	\( T^{32} \) T^32
$73$	\( T^{32} + 8 T^{31} + \cdots + 1 \) T^32 + 8T^31 + 36T^30 + 120T^29 + 330T^28 + 784T^27 + 1652T^26 + 3144T^25 + 5474T^24 + 8824T^23 + 13324T^22 + 18944T^21 + 24864T^20 + 27104T^19 + 19804T^18 + 2344T^17 - 10980T^16 + 9168T^15 + 72764T^14 + 149184T^13 + 182014T^12 + 135696T^11 + 49992T^10 - 10936T^9 - 26227T^8 - 23424T^7 - 14364T^6 - 3024T^5 + 3004T^4 + 1640T^3 + 312T^2 + 24*T + 1
$79$	\( T^{32} - 275 T^{24} + \cdots + 390625 \) T^32 - 275T^24 + 28750T^16 + 62500*T^8 + 390625
$83$	\( T^{32} \) T^32
$89$	\( T^{32} \) T^32
$97$	\( T^{32} - 12 T^{28} + \cdots + 1 \) T^32 - 12T^28 + 140T^24 - 1632T^20 + 40T^18 + 23574T^16 + 70240T^14 + 93152T^12 + 61360T^10 + 13665T^8 - 5280T^6 + 1872T^4 - 80T^2 + 1
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\(n\)	\(730\)	\(965\)
\(\chi(n)\)	\(-\zeta_{80}^{37}\)	\(1\)