LMFDB - Newform orbit 2944.1.p.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2944,1,Mod(689,2944)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2944.689"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2944, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 5, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2944 = 2^{7} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2944.p (of order $8$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.46924739719$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 736)
Projective image:	$D_{24}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{24} + \cdots)$

$q$-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_{24}^{8} - \zeta_{24}^{7}) q^{3} + ( - \zeta_{24}^{4} + \cdots - \zeta_{24}^{2}) q^{9} + (\zeta_{24}^{10} + \zeta_{24}^{5}) q^{13} + \zeta_{24}^{3} q^{23} + \zeta_{24}^{9} q^{25} + (\zeta_{24}^{11} - \zeta_{24}^{10} + \cdots + 1) q^{27}+ \cdots + (\zeta_{24}^{9} + \cdots - \zeta_{24}^{6}) q^{93}+O(q^{100}) $ q + (z^8 - z^7) * q^3 + (-z^4 + z^3 - z^2) * q^9 + (z^10 + z^5) * q^13 + z^3 * q^23 + z^9 * q^25 + (z^11 - z^10 + z^9 + 1) * q^27 + (z^2 + z) * q^29 + (-z^11 + z) * q^31 + (-z^6 + z^5 - z + 1) * q^39 + z^3 * q^41 + (-z^8 - z^4) * q^47 - z^6 * q^49 + (z^6 + z^3) * q^59 + (z^11 - z^10) * q^69 + (z^4 - z^2) * q^71 + (-z^5 - z) * q^73 + (-z^5 + z^4) * q^75 + (z^8 - z^7 + z^6 - z^5 + z^4) * q^81 + (z^10 - z^8) * q^87 + (z^9 - z^8 + z^7 - z^6) * q^93
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 4 q^{9} + 8 q^{27} + 8 q^{39} + 4 q^{71} + 4 q^{75} + 4 q^{87} + 4 q^{93}+O(q^{100}) $ 8 * q - 4 * q^3 - 4 * q^9 + 8 * q^27 + 8 * q^39 + 4 * q^71 + 4 * q^75 + 4 * q^87 + 4 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2944\mathbb{Z}\right)^\times$.

$n$	$645$	$1151$	$2305$
$\chi(n)$	$\zeta_{24}^{9}$	$1$	$-1$

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} $

689.1

−0.258819	+	0.965926i
0.965926	−	0.258819i
−0.965926	−	0.258819i
0.258819	+	0.965926i
−0.965926	+	0.258819i
0.258819	−	0.965926i
−0.258819	−	0.965926i
0.965926	+	0.258819i

−1.46593

0.607206i

1.07313

−

1.07313i

689.2

−0.241181

0.0999004i

−0.658919

0.658919i

1425.1

−0.758819

1.83195i

−2.07313

−

2.07313i

1425.2

0.465926

−

1.12484i

−0.341081

−

0.341081i

2161.1

−0.758819

−

1.83195i

−2.07313

2.07313i

2161.2

0.465926

1.12484i

−0.341081

0.341081i

2897.1

−1.46593

−

0.607206i

1.07313

1.07313i

2897.2

−0.241181

−

0.0999004i

−0.658919

−

0.658919i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	CM by $\Q(\sqrt{-23}) $
32.g	even	8	1	inner
736.p	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2944.1.p.b		8
4.b	odd	2	1		736.1.p.b	✓	8
23.b	odd	2	1	CM	2944.1.p.b		8
32.g	even	8	1	inner	2944.1.p.b		8
32.h	odd	8	1		736.1.p.b	✓	8
92.b	even	2	1		736.1.p.b	✓	8
736.n	even	8	1		736.1.p.b	✓	8
736.p	odd	8	1	inner	2944.1.p.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
736.1.p.b	✓	8	4.b	odd	2	1
736.1.p.b	✓	8	32.h	odd	8	1
736.1.p.b	✓	8	92.b	even	2	1
736.1.p.b	✓	8	736.n	even	8	1
2944.1.p.b		8	1.a	even	1	1	trivial
2944.1.p.b		8	23.b	odd	2	1	CM
2944.1.p.b		8	32.g	even	8	1	inner
2944.1.p.b		8	736.p	odd	8	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 4T_{3}^{7} + 10T_{3}^{6} + 16T_{3}^{5} + 18T_{3}^{4} + 20T_{3}^{3} + 22T_{3}^{2} + 8T_{3} + 1 $ T3^8 + 4*T3^7 + 10*T3^6 + 16*T3^5 + 18*T3^4 + 20*T3^3 + 22*T3^2 + 8*T3 + 1 acting on $S_{1}^{\mathrm{new}}(2944, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 4 T^{7} + \cdots + 1 $ T^8 + 4T^7 + 10T^6 + 16T^5 + 18T^4 + 20T^3 + 22T^2 + 8*T + 1
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} - 2 T^{6} + \cdots + 1 $ T^8 - 2T^6 - 4T^5 + 2T^4 + 16T^3 + 18T^2 + 4T + 1
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$29$	$ T^{8} - 2 T^{6} + \cdots + 1 $ T^8 - 2T^6 - 4T^5 + 2T^4 + 16T^3 + 18T^2 + 4T + 1
$31$	$ (T^{4} - 4 T^{2} + 1)^{2} $ (T^4 - 4*T^2 + 1)^2
$37$	$ T^{8} $ T^8
$41$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$43$	$ T^{8} $ T^8
$47$	$ (T^{2} + 3)^{4} $ (T^2 + 3)^4
$53$	$ T^{8} $ T^8
$59$	$ (T^{4} + 2 T^{2} + 4 T + 2)^{2} $ (T^4 + 2T^2 + 4T + 2)^2
$61$	$ T^{8} $ T^8
$67$	$ T^{8} $ T^8
$71$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$73$	$ (T^{4} + 9)^{2} $ (T^4 + 9)^2
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2944 = 2^{7} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2944.p (of order \(8\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{24}^{8} - \zeta_{24}^{7}) q^{3} + ( - \zeta_{24}^{4} + \cdots - \zeta_{24}^{2}) q^{9} + (\zeta_{24}^{10} + \zeta_{24}^{5}) q^{13} + \zeta_{24}^{3} q^{23} + \zeta_{24}^{9} q^{25} + (\zeta_{24}^{11} - \zeta_{24}^{10} + \cdots + 1) q^{27}+ \cdots + (\zeta_{24}^{9} + \cdots - \zeta_{24}^{6}) q^{93}+O(q^{100}) \) q + (z^8 - z^7) * q^3 + (-z^4 + z^3 - z^2) * q^9 + (z^10 + z^5) * q^13 + z^3 * q^23 + z^9 * q^25 + (z^11 - z^10 + z^9 + 1) * q^27 + (z^2 + z) * q^29 + (-z^11 + z) * q^31 + (-z^6 + z^5 - z + 1) * q^39 + z^3 * q^41 + (-z^8 - z^4) * q^47 - z^6 * q^49 + (z^6 + z^3) * q^59 + (z^11 - z^10) * q^69 + (z^4 - z^2) * q^71 + (-z^5 - z) * q^73 + (-z^5 + z^4) * q^75 + (z^8 - z^7 + z^6 - z^5 + z^4) * q^81 + (z^10 - z^8) * q^87 + (z^9 - z^8 + z^7 - z^6) * q^93
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 4 q^{9} + 8 q^{27} + 8 q^{39} + 4 q^{71} + 4 q^{75} + 4 q^{87} + 4 q^{93}+O(q^{100}) \) 8 * q - 4 * q^3 - 4 * q^9 + 8 * q^27 + 8 * q^39 + 4 * q^71 + 4 * q^75 + 4 * q^87 + 4 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more