LMFDB - Newform orbit 884.1.ct.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [884,1,Mod(7,884)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(884, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 44, 33]))

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

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

chi := DirichletCharacter("884.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 884 = 2^{2} \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	884.ct (of order $48$, 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:	$0.441173471168$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{48}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{48} - \cdots)$

$q$-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_{48}^{17} q^{2} - \zeta_{48}^{10} q^{4} + ( - \zeta_{48}^{19} + \zeta_{48}^{2}) q^{5} + \zeta_{48}^{3} q^{8} + \zeta_{48}^{13} q^{9} +O(q^{10}) $ q + z^17 * q^2 - z^10 * q^4 + (-z^19 + z^2) * q^5 + z^3 * q^8 + z^13 * q^9	$ q + \zeta_{48}^{17} q^{2} - \zeta_{48}^{10} q^{4} + ( - \zeta_{48}^{19} + \zeta_{48}^{2}) q^{5} + \zeta_{48}^{3} q^{8} + \zeta_{48}^{13} q^{9} + (\zeta_{48}^{19} + \zeta_{48}^{12}) q^{10} + \zeta_{48}^{23} q^{13} + \zeta_{48}^{20} q^{16} - \zeta_{48}^{18} q^{17} - \zeta_{48}^{6} q^{18} + ( - \zeta_{48}^{12} - \zeta_{48}^{5}) q^{20} + ( - \zeta_{48}^{21} + \cdots + \zeta_{48}^{4}) q^{25} + \cdots + \zeta_{48}^{4} q^{98} +O(q^{100}) $ q + z^17 * q^2 - z^10 * q^4 + (-z^19 + z^2) * q^5 + z^3 * q^8 + z^13 * q^9 + (z^19 + z^12) * q^10 + z^23 * q^13 + z^20 * q^16 - z^18 * q^17 - z^6 * q^18 + (-z^12 - z^5) * q^20 + (-z^21 - z^14 + z^4) * q^25 - z^16 * q^26 + (z - 1) * q^29 - z^13 * q^32 + z^11 * q^34 - z^23 * q^36 + (-z^15 + z^10) * q^37 + (-z^22 + z^5) * q^40 + (z^16 - z^3) * q^41 + (z^15 + z^8) * q^45 - z^11 * q^49 + (z^21 + z^14 + z^7) * q^50 + z^9 * q^52 + (z^22 - z^8) * q^53 + (z^18 - z^17) * q^58 + (z^14 + z^9) * q^61 + z^6 * q^64 + (z^18 - z) * q^65 - z^4 * q^68 + z^16 * q^72 + (-z^17 - z^4) * q^73 + (z^8 - z^3) * q^74 + (z^22 + z^15) * q^80 - z^2 * q^81 + (-z^20 - z^9) * q^82 + (-z^20 - z^13) * q^85 + (z^11 + z^5) * q^89 + (-z^8 - z) * q^90 + (-z^22 + z^7) * q^97 + z^4 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 8 q^{26} - 16 q^{29} - 8 q^{41} + 8 q^{45} - 8 q^{53} - 8 q^{72} + 8 q^{74} - 8 q^{90}+O(q^{100}) $ 16 * q + 8 * q^26 - 16 * q^29 - 8 * q^41 + 8 * q^45 - 8 * q^53 - 8 * q^72 + 8 * q^74 - 8 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/884\mathbb{Z}\right)^\times$.

$n$	$105$	$443$	$613$
$\chi(n)$	$-\zeta_{48}^{21}$	$-1$	$-\zeta_{48}^{4}$

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} $

7.1

0.793353	+	0.608761i
−0.991445	−	0.130526i
0.991445	−	0.130526i
−0.793353	−	0.608761i
0.130526	+	0.991445i
0.608761	−	0.793353i
−0.608761	−	0.793353i
0.793353	−	0.608761i
−0.130526	−	0.991445i
0.991445	+	0.130526i
−0.991445	+	0.130526i
0.130526	−	0.991445i
−0.793353	+	0.608761i
−0.608761	+	0.793353i
−0.130526	+	0.991445i
0.608761	+	0.793353i

0.130526

−

0.991445i

−0.965926

−

0.258819i

−0.732626

1.09645i

−0.382683

0.923880i

−0.608761

0.793353i

0.991445

0.869474i

11.1

0.608761

−

0.793353i

−0.258819

−

0.965926i

0.172572

0.867580i

−0.923880

−

0.382683i

0.130526

−

0.991445i

0.793353

0.391239i

71.1

−0.608761

−

0.793353i

−0.258819

0.965926i

1.75928

0.349942i

0.923880

−

0.382683i

−0.130526

−

0.991445i

−0.793353

−

1.60876i

163.1

−0.130526

0.991445i

−0.965926

−

0.258819i

1.25026

0.835400i

0.382683

−

0.923880i

0.608761

−

0.793353i

−0.991445

1.13053i

175.1

0.793353

−

0.608761i

0.258819

−

0.965926i

−0.357164

−

0.534534i

−0.382683

−

0.923880i

0.991445

−

0.130526i

−0.608761

−

0.206647i

215.1

−0.991445

−

0.130526i

0.965926

0.258819i

−0.389345

−

1.95737i

−0.923880

−

0.382683i

0.793353

0.608761i

0.130526

1.99144i

275.1

0.991445

−

0.130526i

0.965926

−

0.258819i

−0.128293

−

0.0255190i

0.923880

−

0.382683i

−0.793353

0.608761i

−0.130526

−

0.00855514i

379.1

0.130526

0.991445i

−0.965926

0.258819i

−0.732626

−

1.09645i

−0.382683

−

0.923880i

−0.608761

−

0.793353i

0.991445

−

0.869474i

539.1

−0.793353

0.608761i

0.258819

−

0.965926i

−1.57469

1.05217i

0.382683

0.923880i

−0.991445

0.130526i

0.608761

−

1.79335i

635.1

−0.608761

0.793353i

−0.258819

−

0.965926i

1.75928

−

0.349942i

0.923880

0.382683i

−0.130526

0.991445i

−0.793353

1.60876i

643.1

0.608761

0.793353i

−0.258819

0.965926i

0.172572

−

0.867580i

−0.923880

0.382683i

0.130526

0.991445i

0.793353

−

0.391239i

687.1

0.793353

0.608761i

0.258819

0.965926i

−0.357164

0.534534i

−0.382683

0.923880i

0.991445

0.130526i

−0.608761

0.206647i

743.1

−0.130526

−

0.991445i

−0.965926

0.258819i

1.25026

−

0.835400i

0.382683

0.923880i

0.608761

0.793353i

−0.991445

−

1.13053i

839.1

0.991445

0.130526i

0.965926

0.258819i

−0.128293

0.0255190i

0.923880

0.382683i

−0.793353

−

0.608761i

−0.130526

0.00855514i

843.1

−0.793353

−

0.608761i

0.258819

0.965926i

−1.57469

−

1.05217i

0.382683

−

0.923880i

−0.991445

−

0.130526i

0.608761

1.79335i

847.1

−0.991445

0.130526i

0.965926

−

0.258819i

−0.389345

1.95737i

−0.923880

0.382683i

0.793353

−

0.608761i

0.130526

−

1.99144i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
221.bi	even	48	1	inner
884.ct	odd	48	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	884.1.ct.a	yes	16
4.b	odd	2	1	CM	884.1.ct.a	yes	16
13.f	odd	12	1		884.1.cn.a	✓	16
17.e	odd	16	1		884.1.cn.a	✓	16
52.l	even	12	1		884.1.cn.a	✓	16
68.i	even	16	1		884.1.cn.a	✓	16
221.bi	even	48	1	inner	884.1.ct.a	yes	16
884.ct	odd	48	1	inner	884.1.ct.a	yes	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
884.1.cn.a	✓	16	13.f	odd	12	1
884.1.cn.a	✓	16	17.e	odd	16	1
884.1.cn.a	✓	16	52.l	even	12	1
884.1.cn.a	✓	16	68.i	even	16	1
884.1.ct.a	yes	16	1.a	even	1	1	trivial
884.1.ct.a	yes	16	4.b	odd	2	1	CM
884.1.ct.a	yes	16	221.bi	even	48	1	inner
884.1.ct.a	yes	16	884.ct	odd	48	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(884, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - T^{8} + 1 $ T^16 - T^8 + 1
$3$	$ T^{16} $ T^16
$5$	$ T^{16} - 8 T^{13} + \cdots + 1 $ T^16 - 8T^13 - 2T^12 + 52T^10 - 32T^9 + 2T^8 - 96T^7 + 112T^6 + 24T^5 + 146T^4 + 96T^3 + 76T^2 + 16T + 1
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} - T^{8} + 1 $ T^16 - T^8 + 1
$17$	$ (T^{4} + 1)^{4} $ (T^4 + 1)^4
$19$	$ T^{16} $ T^16
$23$	$ T^{16} $ T^16
$29$	$ T^{16} + 16 T^{15} + \cdots + 1 $ T^16 + 16T^15 + 120T^14 + 560T^13 + 1820T^12 + 4368T^11 + 8008T^10 + 11440T^9 + 12869T^8 + 11432T^7 + 7980T^6 + 4312T^5 + 1750T^4 + 504T^3 + 92T^2 + 8*T + 1
$31$	$ T^{16} $ T^16
$37$	$ T^{16} - 8 T^{13} + \cdots + 1 $ T^16 - 8T^13 - 2T^12 + 52T^10 + 40T^9 + 5T^8 - 96T^7 + 40T^6 + 140T^4 - 192T^3 + 76T^2 - 8*T + 1
$41$	$ T^{16} + 8 T^{15} + \cdots + 1 $ T^16 + 8T^15 + 36T^14 + 112T^13 + 266T^12 + 504T^11 + 784T^10 + 1016T^9 + 1109T^8 + 1024T^7 + 756T^6 + 392T^5 + 196T^4 + 168T^3 + 92T^2 + 16*T + 1
$43$	$ T^{16} $ T^16
$47$	$ T^{16} $ T^16
$53$	$ (T^{8} + 4 T^{7} + 10 T^{6} + \cdots + 1)^{2} $ (T^8 + 4T^7 + 10T^6 + 16T^5 + 18T^4 + 20T^3 + 22T^2 + 8*T + 1)^2
$59$	$ T^{16} $ T^16
$61$	$ T^{16} + 8 T^{13} + \cdots + 1 $ T^16 + 8T^13 - 2T^12 + 52T^10 - 40T^9 + 5T^8 + 96T^7 + 40T^6 + 140T^4 + 192T^3 + 76T^2 + 8*T + 1
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} - 4 T^{14} + \cdots + 1 $ T^16 - 4T^14 + 10T^12 - 16T^11 - 16T^10 + 24T^9 + 18T^8 + 112T^7 + 44T^6 - 112T^5 + 86T^4 - 16T^3 + 56T^2 + 8*T + 1
$79$	$ T^{16} $ T^16
$83$	$ T^{16} $ T^16
$89$	$ (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} $ (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$97$	$ T^{16} - 2 T^{12} + \cdots + 4 $ T^16 - 2T^12 - 16T^11 + 40T^10 - 8T^9 + 2T^8 - 32T^7 + 216T^6 - 288T^5 + 268T^4 - 96T^3 + 24T^2 - 16T + 4
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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:	\( N \)	\(=\)	\( 884 = 2^{2} \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	884.ct (of order \(48\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{48}^{17} q^{2} - \zeta_{48}^{10} q^{4} + ( - \zeta_{48}^{19} + \zeta_{48}^{2}) q^{5} + \zeta_{48}^{3} q^{8} + \zeta_{48}^{13} q^{9} +O(q^{10}) \) q + z^17 * q^2 - z^10 * q^4 + (-z^19 + z^2) * q^5 + z^3 * q^8 + z^13 * q^9	\( q + \zeta_{48}^{17} q^{2} - \zeta_{48}^{10} q^{4} + ( - \zeta_{48}^{19} + \zeta_{48}^{2}) q^{5} + \zeta_{48}^{3} q^{8} + \zeta_{48}^{13} q^{9} + (\zeta_{48}^{19} + \zeta_{48}^{12}) q^{10} + \zeta_{48}^{23} q^{13} + \zeta_{48}^{20} q^{16} - \zeta_{48}^{18} q^{17} - \zeta_{48}^{6} q^{18} + ( - \zeta_{48}^{12} - \zeta_{48}^{5}) q^{20} + ( - \zeta_{48}^{21} + \cdots + \zeta_{48}^{4}) q^{25} + \cdots + \zeta_{48}^{4} q^{98} +O(q^{100}) \) q + z^17 * q^2 - z^10 * q^4 + (-z^19 + z^2) * q^5 + z^3 * q^8 + z^13 * q^9 + (z^19 + z^12) * q^10 + z^23 * q^13 + z^20 * q^16 - z^18 * q^17 - z^6 * q^18 + (-z^12 - z^5) * q^20 + (-z^21 - z^14 + z^4) * q^25 - z^16 * q^26 + (z - 1) * q^29 - z^13 * q^32 + z^11 * q^34 - z^23 * q^36 + (-z^15 + z^10) * q^37 + (-z^22 + z^5) * q^40 + (z^16 - z^3) * q^41 + (z^15 + z^8) * q^45 - z^11 * q^49 + (z^21 + z^14 + z^7) * q^50 + z^9 * q^52 + (z^22 - z^8) * q^53 + (z^18 - z^17) * q^58 + (z^14 + z^9) * q^61 + z^6 * q^64 + (z^18 - z) * q^65 - z^4 * q^68 + z^16 * q^72 + (-z^17 - z^4) * q^73 + (z^8 - z^3) * q^74 + (z^22 + z^15) * q^80 - z^2 * q^81 + (-z^20 - z^9) * q^82 + (-z^20 - z^13) * q^85 + (z^11 + z^5) * q^89 + (-z^8 - z) * q^90 + (-z^22 + z^7) * q^97 + z^4 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 8 q^{26} - 16 q^{29} - 8 q^{41} + 8 q^{45} - 8 q^{53} - 8 q^{72} + 8 q^{74} - 8 q^{90}+O(q^{100}) \) 16 * q + 8 * q^26 - 16 * q^29 - 8 * q^41 + 8 * q^45 - 8 * q^53 - 8 * q^72 + 8 * q^74 - 8 * q^90

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	884.1.ct.a

Level	$884$
Weight	$1$
Character orbit	884.ct
Analytic conductor	$0.441$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{48}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.441173471168\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{48})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{8} + 1 \) x^16 - x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{48}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

\(n\)	\(105\)	\(443\)	\(613\)
\(\chi(n)\)	\(-\zeta_{48}^{21}\)	\(-1\)	\(-\zeta_{48}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
221.bi	even	48	1	inner
884.ct	odd	48	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$3$	\( T^{16} \) T^16
$5$	\( T^{16} - 8 T^{13} + \cdots + 1 \) T^16 - 8T^13 - 2T^12 + 52T^10 - 32T^9 + 2T^8 - 96T^7 + 112T^6 + 24T^5 + 146T^4 + 96T^3 + 76T^2 + 16T + 1
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$17$	\( (T^{4} + 1)^{4} \) (T^4 + 1)^4
$19$	\( T^{16} \) T^16
$23$	\( T^{16} \) T^16
$29$	\( T^{16} + 16 T^{15} + \cdots + 1 \) T^16 + 16T^15 + 120T^14 + 560T^13 + 1820T^12 + 4368T^11 + 8008T^10 + 11440T^9 + 12869T^8 + 11432T^7 + 7980T^6 + 4312T^5 + 1750T^4 + 504T^3 + 92T^2 + 8*T + 1
$31$	\( T^{16} \) T^16
$37$	\( T^{16} - 8 T^{13} + \cdots + 1 \) T^16 - 8T^13 - 2T^12 + 52T^10 + 40T^9 + 5T^8 - 96T^7 + 40T^6 + 140T^4 - 192T^3 + 76T^2 - 8*T + 1
$41$	\( T^{16} + 8 T^{15} + \cdots + 1 \) T^16 + 8T^15 + 36T^14 + 112T^13 + 266T^12 + 504T^11 + 784T^10 + 1016T^9 + 1109T^8 + 1024T^7 + 756T^6 + 392T^5 + 196T^4 + 168T^3 + 92T^2 + 16*T + 1
$43$	\( T^{16} \) T^16
$47$	\( T^{16} \) T^16
$53$	\( (T^{8} + 4 T^{7} + 10 T^{6} + \cdots + 1)^{2} \) (T^8 + 4T^7 + 10T^6 + 16T^5 + 18T^4 + 20T^3 + 22T^2 + 8*T + 1)^2
$59$	\( T^{16} \) T^16
$61$	\( T^{16} + 8 T^{13} + \cdots + 1 \) T^16 + 8T^13 - 2T^12 + 52T^10 - 40T^9 + 5T^8 + 96T^7 + 40T^6 + 140T^4 + 192T^3 + 76T^2 + 8*T + 1
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} - 4 T^{14} + \cdots + 1 \) T^16 - 4T^14 + 10T^12 - 16T^11 - 16T^10 + 24T^9 + 18T^8 + 112T^7 + 44T^6 - 112T^5 + 86T^4 - 16T^3 + 56T^2 + 8*T + 1
$79$	\( T^{16} \) T^16
$83$	\( T^{16} \) T^16
$89$	\( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) (T^8 + 4T^6 + 14T^4 + 8*T^2 + 4)^2
$97$	\( T^{16} - 2 T^{12} + \cdots + 4 \) T^16 - 2T^12 - 16T^11 + 40T^10 - 8T^9 + 2T^8 - 32T^7 + 216T^6 - 288T^5 + 268T^4 - 96T^3 + 24T^2 - 16T + 4
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