LMFDB - Newform orbit 884.1.cn.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [884,1,Mod(63,884)]

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, 28, 39]))

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

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

chi := DirichletCharacter("884.63");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 884 = 2^{2} \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	884.cn (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}^{22} - \zeta_{48}^{11}) q^{5} + \zeta_{48}^{3} q^{8} - \zeta_{48} q^{9} +O(q^{10}) $ q + z^17 * q^2 - z^10 * q^4 + (-z^22 - z^11) * q^5 + z^3 * q^8 - z * q^9	$ q + \zeta_{48}^{17} q^{2} - \zeta_{48}^{10} q^{4} + ( - \zeta_{48}^{22} - \zeta_{48}^{11}) q^{5} + \zeta_{48}^{3} q^{8} - \zeta_{48} q^{9} + (\zeta_{48}^{15} + \zeta_{48}^{4}) q^{10} - \zeta_{48}^{17} q^{13} + \zeta_{48}^{20} q^{16} - \zeta_{48}^{2} q^{17} - \zeta_{48}^{18} q^{18} + (\zeta_{48}^{21} - \zeta_{48}^{8}) q^{20} + (\zeta_{48}^{22} + \cdots - \zeta_{48}^{9}) q^{25} + \cdots + \zeta_{48}^{16} q^{98} +O(q^{100}) $ q + z^17 * q^2 - z^10 * q^4 + (-z^22 - z^11) * q^5 + z^3 * q^8 - z * q^9 + (z^15 + z^4) * q^10 - z^17 * q^13 + z^20 * q^16 - z^2 * q^17 - z^18 * q^18 + (z^21 - z^8) * q^20 + (z^22 - z^20 - z^9) * q^25 + z^10 * q^26 + (-z^21 - z^16) * q^29 - z^13 * q^32 - z^19 * q^34 + z^11 * q^36 + (z^7 - z^6) * q^37 + (-z^14 + z) * q^40 + (z^19 - z^12) * q^41 + (z^23 + z^12) * q^45 - z^23 * q^49 + (-z^15 + z^13 + z^2) * q^50 - z^3 * q^52 + (z^16 + z^14) * q^53 + (z^14 + z^9) * q^58 + (z^6 + z^5) * q^61 + z^6 * q^64 + (-z^15 - z^4) * q^65 + z^12 * q^68 - z^4 * q^72 + (z^8 + z) * q^73 + (-z^23 - 1) * q^74 + (z^18 + z^7) * q^80 + z^2 * q^81 + (-z^12 + z^5) * q^82 + (z^13 - 1) * q^85 + (z^11 - z^5) * q^89 + (-z^16 - z^5) * q^90 + (z^10 - z^7) * q^97 + z^16 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{20} + 8 q^{29} - 8 q^{53} + 8 q^{73} - 16 q^{74} - 16 q^{85} + 8 q^{90} - 8 q^{98}+O(q^{100}) $ 16 * q - 8 * q^20 + 8 * q^29 - 8 * q^53 + 8 * q^73 - 16 * q^74 - 16 * q^85 + 8 * q^90 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$105$	$443$	$613$
$\chi(n)$	$\zeta_{48}^{9}$	$-1$	$\zeta_{48}^{4}$

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} $

63.1

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

0.991445

−

0.130526i

0.965926

−

0.258819i

−1.05217

−

1.57469i

0.923880

−

0.382683i

0.608761

0.793353i

−1.24871

−

1.42388i

167.1

−0.130526

−

0.991445i

−0.965926

0.258819i

0.867580

0.172572i

0.382683

0.923880i

0.793353

−

0.608761i

0.0578541

−

0.882683i

227.1

0.130526

−

0.991445i

−0.965926

−

0.258819i

−0.349942

−

1.75928i

−0.382683

0.923880i

−0.793353

−

0.608761i

−1.78990

0.117317i

267.1

−0.608761

−

0.793353i

−0.258819

0.965926i

0.835400

1.25026i

0.923880

−

0.382683i

−0.991445

0.130526i

0.483342

−

1.42388i

279.1

−0.991445

−

0.130526i

0.965926

0.258819i

0.534534

0.357164i

−0.923880

−

0.382683i

−0.608761

0.793353i

−0.483342

−

0.423880i

371.1

−0.793353

0.608761i

0.258819

−

0.965926i

−1.95737

−

0.389345i

0.382683

0.923880i

0.130526

0.991445i

1.78990

−

0.882683i

431.1

0.793353

0.608761i

0.258819

0.965926i

0.0255190

0.128293i

−0.382683

0.923880i

−0.130526

0.991445i

−0.0578541

0.117317i

435.1

0.991445

0.130526i

0.965926

0.258819i

−1.05217

1.57469i

0.923880

0.382683i

0.608761

−

0.793353i

−1.24871

1.42388i

479.1

0.130526

0.991445i

−0.965926

0.258819i

−0.349942

1.75928i

−0.382683

−

0.923880i

−0.793353

0.608761i

−1.78990

−

0.117317i

483.1

0.608761

−

0.793353i

−0.258819

−

0.965926i

1.09645

0.732626i

−0.923880

−

0.382683i

0.991445

0.130526i

1.24871

−

0.423880i

487.1

−0.130526

0.991445i

−0.965926

−

0.258819i

0.867580

−

0.172572i

0.382683

−

0.923880i

0.793353

0.608761i

0.0578541

0.882683i

583.1

−0.991445

0.130526i

0.965926

−

0.258819i

0.534534

−

0.357164i

−0.923880

0.382683i

−0.608761

−

0.793353i

−0.483342

0.423880i

639.1

−0.608761

0.793353i

−0.258819

−

0.965926i

0.835400

−

1.25026i

0.923880

0.382683i

−0.991445

−

0.130526i

0.483342

1.42388i

683.1

0.793353

−

0.608761i

0.258819

−

0.965926i

0.0255190

−

0.128293i

−0.382683

−

0.923880i

−0.130526

−

0.991445i

−0.0578541

−

0.117317i

691.1

−0.793353

−

0.608761i

0.258819

0.965926i

−1.95737

0.389345i

0.382683

−

0.923880i

0.130526

−

0.991445i

1.78990

0.882683i

787.1

0.608761

0.793353i

−0.258819

0.965926i

1.09645

−

0.732626i

−0.923880

0.382683i

0.991445

−

0.130526i

1.24871

0.423880i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
221.bh	even	48	1	inner
884.cn	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.cn.a	✓	16
4.b	odd	2	1	CM	884.1.cn.a	✓	16
13.f	odd	12	1		884.1.ct.a	yes	16
17.e	odd	16	1		884.1.ct.a	yes	16
52.l	even	12	1		884.1.ct.a	yes	16
68.i	even	16	1		884.1.ct.a	yes	16
221.bh	even	48	1	inner	884.1.cn.a	✓	16
884.cn	odd	48	1	inner	884.1.cn.a	✓	16

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

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} - 2 T^{12} + \cdots + 1 $ T^16 - 2T^12 + 16T^11 - 20T^10 + 8T^9 + 2T^8 - 64T^7 + 336T^6 - 672T^5 + 658T^4 - 336T^3 + 84T^2 - 8T + 1
$7$	$ T^{16} $ T^16
$11$	$ T^{16} $ T^16
$13$	$ T^{16} - T^{8} + 1 $ T^16 - T^8 + 1
$17$	$ (T^{8} - T^{4} + 1)^{2} $ (T^8 - T^4 + 1)^2
$19$	$ T^{16} $ T^16
$23$	$ T^{16} $ T^16
$29$	$ 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
$31$	$ T^{16} $ T^16
$37$	$ T^{16} + 4 T^{12} + \cdots + 1 $ T^16 + 4T^12 - 16T^11 - 20T^10 + 16T^9 + 5T^8 - 32T^7 + 216T^6 - 288T^5 + 274T^4 - 144T^3 + 24T^2 + 8T + 1
$41$	$ T^{16} + 8 T^{14} + \cdots + 1 $ T^16 + 8T^14 + 28T^12 + 8T^11 + 56T^10 + 24T^9 + 69T^8 + 16T^7 + 116T^6 - 16T^5 - 106T^4 - 16T^3 + 68T^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 + 4T^12 + 52T^10 + 8T^9 + 5T^8 + 96T^7 - 152T^6 - 24T^5 + 146T^4 - 96T^3 + 40T^2 + 8T + 1
$67$	$ T^{16} $ T^16
$71$	$ T^{16} $ T^16
$73$	$ T^{16} - 8 T^{15} + \cdots + 1 $ T^16 - 8T^15 + 36T^14 - 112T^13 + 266T^12 - 504T^11 + 784T^10 - 1016T^9 + 1106T^8 - 1000T^7 + 756T^6 - 560T^5 + 406T^4 - 168T^3 + 8T^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} + 16 T^{13} + \cdots + 4 $ T^16 + 16T^13 - 2T^12 + 88T^10 - 8T^9 + 2T^8 + 192T^7 + 40T^6 - 48T^5 + 140T^4 + 96T^3 + 40T^2 + 16T + 4
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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 884 = 2^{2} \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	884.cn (of order \(48\), degree \(16\), minimal)

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

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.cn.a

Level	$884$
Weight	$1$
Character orbit	884.cn
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 \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

\(n\)	\(105\)	\(443\)	\(613\)
\(\chi(n)\)	\(\zeta_{48}^{9}\)	\(-1\)	\(\zeta_{48}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 63.1
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.bh	even	48	1	inner
884.cn	odd	48	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$3$	\( T^{16} \) T^16
$5$	\( T^{16} - 2 T^{12} + \cdots + 1 \) T^16 - 2T^12 + 16T^11 - 20T^10 + 8T^9 + 2T^8 - 64T^7 + 336T^6 - 672T^5 + 658T^4 - 336T^3 + 84T^2 - 8T + 1
$7$	\( T^{16} \) T^16
$11$	\( T^{16} \) T^16
$13$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$17$	\( (T^{8} - T^{4} + 1)^{2} \) (T^8 - T^4 + 1)^2
$19$	\( T^{16} \) T^16
$23$	\( T^{16} \) T^16
$29$	\( T^{16} - 8 T^{15} + \cdots + 1 \) 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
$31$	\( T^{16} \) T^16
$37$	\( T^{16} + 4 T^{12} + \cdots + 1 \) T^16 + 4T^12 - 16T^11 - 20T^10 + 16T^9 + 5T^8 - 32T^7 + 216T^6 - 288T^5 + 274T^4 - 144T^3 + 24T^2 + 8T + 1
$41$	\( T^{16} + 8 T^{14} + \cdots + 1 \) T^16 + 8T^14 + 28T^12 + 8T^11 + 56T^10 + 24T^9 + 69T^8 + 16T^7 + 116T^6 - 16T^5 - 106T^4 - 16T^3 + 68T^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 + 4T^12 + 52T^10 + 8T^9 + 5T^8 + 96T^7 - 152T^6 - 24T^5 + 146T^4 - 96T^3 + 40T^2 + 8T + 1
$67$	\( T^{16} \) T^16
$71$	\( T^{16} \) T^16
$73$	\( T^{16} - 8 T^{15} + \cdots + 1 \) T^16 - 8T^15 + 36T^14 - 112T^13 + 266T^12 - 504T^11 + 784T^10 - 1016T^9 + 1106T^8 - 1000T^7 + 756T^6 - 560T^5 + 406T^4 - 168T^3 + 8T^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
$97$	\( T^{16} + 16 T^{13} + \cdots + 4 \) T^16 + 16T^13 - 2T^12 + 88T^10 - 8T^9 + 2T^8 + 192T^7 + 40T^6 - 48T^5 + 140T^4 + 96T^3 + 40T^2 + 16T + 4
show more
show less