LMFDB - Newform orbit 1376.1.be.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1376,1,Mod(47,1376)] 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("1376.47"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1376, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 7, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1376 = 2^{5} \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1376.be (of order $14$, degree $6$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$0.686713457383$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 344)
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.3236537881088.1

$q$-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_{14}^{4} + \zeta_{14}) q^{3} + ( - \zeta_{14}^{5} + \zeta_{14}^{2} - \zeta_{14}) q^{9} + ( - \zeta_{14}^{6} - \zeta_{14}^{4}) q^{11} + (\zeta_{14}^{6} + \zeta_{14}^{2}) q^{17} + ( - \zeta_{14}^{4} - 1) q^{19}+ \cdots + ( - \zeta_{14}^{6} + \zeta_{14}^{5} + \cdots - 1) q^{99}+O(q^{100}) $ q + (-z^4 + z) * q^3 + (-z^5 + z^2 - z) * q^9 + (-z^6 - z^4) * q^11 + (z^6 + z^2) * q^17 + (-z^4 - 1) * q^19 + z^6 * q^25 + (-z^6 + z^5 + z^3 - z^2) * q^27 + (-z^5 - z^3 - z + 1) * q^33 + (-z^5 + z^4) * q^41 + z * q^43 + q^49 + (-z^6 + 2z^3 - 1) q^51 + (-z^5 + z^4 - 2z) q^57 + (z^5 + z^3) * q^59 - 2z^2 q^67 - 2z^3 q^73 + (z^3 - 1) * q^75 + (z^6 + z^4 - z^3 + z^2 + 1) * q^81 + (z^5 - 1) * q^83 + (-z^3 + z^2) * q^89 + (z^6 + z^4) * q^97 + (-z^6 + z^5 - z^4 - z^2 + z - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 3 q^{9} + 2 q^{11} - 2 q^{17} - 5 q^{19} - q^{25} + 4 q^{27} + 3 q^{33} - 2 q^{41} + q^{43} + 6 q^{49} - 3 q^{51} - 4 q^{57} + 2 q^{59} + 2 q^{67} - 2 q^{73} - 5 q^{75} + 2 q^{81} - 5 q^{83}+ \cdots - q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 3 * q^9 + 2 * q^11 - 2 * q^17 - 5 * q^19 - q^25 + 4 * q^27 + 3 * q^33 - 2 * q^41 + q^43 + 6 * q^49 - 3 * q^51 - 4 * q^57 + 2 * q^59 + 2 * q^67 - 2 * q^73 - 5 * q^75 + 2 * q^81 - 5 * q^83 - 2 * q^89 - 2 * q^97 - q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1376\mathbb{Z}\right)^\times$.

$n$	$517$	$1119$	$1121$
$\chi(n)$	$-1$	$-1$	$-\zeta_{14}^{5}$

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} $

47.1

−0.623490	+	0.781831i
0.222521	−	0.974928i
−0.623490	−	0.781831i
0.900969	+	0.433884i
0.900969	−	0.433884i
0.222521	+	0.974928i

0.277479

0.347948i

0.178448

−

0.781831i

207.1

−0.400969

−

1.75676i

−2.02446

0.974928i

527.1

0.277479

−

0.347948i

0.178448

0.781831i

623.1

1.12349

−

0.541044i

0.346011

−

0.433884i

815.1

1.12349

0.541044i

0.346011

0.433884i

1263.1

−0.400969

1.75676i

−2.02446

−

0.974928i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
43.e	even	7	1	inner
344.s	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1376.1.be.a		6
4.b	odd	2	1		344.1.s.a	✓	6
8.b	even	2	1		344.1.s.a	✓	6
8.d	odd	2	1	CM	1376.1.be.a		6
12.b	even	2	1		3096.1.dm.a		6
24.h	odd	2	1		3096.1.dm.a		6
43.e	even	7	1	inner	1376.1.be.a		6
172.k	odd	14	1		344.1.s.a	✓	6
344.s	odd	14	1	inner	1376.1.be.a		6
344.x	even	14	1		344.1.s.a	✓	6
516.v	even	14	1		3096.1.dm.a		6
1032.bn	odd	14	1		3096.1.dm.a		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
344.1.s.a	✓	6	4.b	odd	2	1
344.1.s.a	✓	6	8.b	even	2	1
344.1.s.a	✓	6	172.k	odd	14	1
344.1.s.a	✓	6	344.x	even	14	1
1376.1.be.a		6	1.a	even	1	1	trivial
1376.1.be.a		6	8.d	odd	2	1	CM
1376.1.be.a		6	43.e	even	7	1	inner
1376.1.be.a		6	344.s	odd	14	1	inner
3096.1.dm.a		6	12.b	even	2	1
3096.1.dm.a		6	24.h	odd	2	1
3096.1.dm.a		6	516.v	even	14	1
3096.1.dm.a		6	1032.bn	odd	14	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1376, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$5$	$ T^{6} $ T^6
$7$	$ T^{6} $ T^6
$11$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - T^3 + 2T^2 + 3T + 1
$13$	$ T^{6} $ T^6
$17$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$19$	$ T^{6} + 5 T^{5} + \cdots + 1 $ T^6 + 5T^5 + 11T^4 + 13T^3 + 9T^2 + 3*T + 1
$23$	$ T^{6} $ T^6
$29$	$ T^{6} $ T^6
$31$	$ T^{6} $ T^6
$37$	$ T^{6} $ T^6
$41$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$43$	$ T^{6} - T^{5} + T^{4} + \cdots + 1 $ T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$47$	$ T^{6} $ T^6
$53$	$ T^{6} $ T^6
$59$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$61$	$ T^{6} $ T^6
$67$	$ T^{6} - 2 T^{5} + \cdots + 64 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 16T^2 - 32*T + 64
$71$	$ T^{6} $ T^6
$73$	$ T^{6} + 2 T^{5} + \cdots + 64 $ T^6 + 2T^5 + 4T^4 + 8T^3 + 16T^2 + 32*T + 64
$79$	$ T^{6} $ T^6
$83$	$ T^{6} + 5 T^{5} + \cdots + 1 $ T^6 + 5T^5 + 11T^4 + 13T^3 + 9T^2 + 3*T + 1
$89$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$97$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
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

Level:	\( N \)	\(=\)	\( 1376 = 2^{5} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1376.be (of order \(14\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{14}^{4} + \zeta_{14}) q^{3} + ( - \zeta_{14}^{5} + \zeta_{14}^{2} - \zeta_{14}) q^{9} + ( - \zeta_{14}^{6} - \zeta_{14}^{4}) q^{11} + (\zeta_{14}^{6} + \zeta_{14}^{2}) q^{17} + ( - \zeta_{14}^{4} - 1) q^{19}+ \cdots + ( - \zeta_{14}^{6} + \zeta_{14}^{5} + \cdots - 1) q^{99}+O(q^{100}) \) q + (-z^4 + z) * q^3 + (-z^5 + z^2 - z) * q^9 + (-z^6 - z^4) * q^11 + (z^6 + z^2) * q^17 + (-z^4 - 1) * q^19 + z^6 * q^25 + (-z^6 + z^5 + z^3 - z^2) * q^27 + (-z^5 - z^3 - z + 1) * q^33 + (-z^5 + z^4) * q^41 + z * q^43 + q^49 + (-z^6 + 2z^3 - 1) q^51 + (-z^5 + z^4 - 2z) q^57 + (z^5 + z^3) * q^59 - 2z^2 q^67 - 2z^3 q^73 + (z^3 - 1) * q^75 + (z^6 + z^4 - z^3 + z^2 + 1) * q^81 + (z^5 - 1) * q^83 + (-z^3 + z^2) * q^89 + (z^6 + z^4) * q^97 + (-z^6 + z^5 - z^4 - z^2 + z - 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 3 q^{9} + 2 q^{11} - 2 q^{17} - 5 q^{19} - q^{25} + 4 q^{27} + 3 q^{33} - 2 q^{41} + q^{43} + 6 q^{49} - 3 q^{51} - 4 q^{57} + 2 q^{59} + 2 q^{67} - 2 q^{73} - 5 q^{75} + 2 q^{81} - 5 q^{83}+ \cdots - q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 3 * q^9 + 2 * q^11 - 2 * q^17 - 5 * q^19 - q^25 + 4 * q^27 + 3 * q^33 - 2 * q^41 + q^43 + 6 * q^49 - 3 * q^51 - 4 * q^57 + 2 * q^59 + 2 * q^67 - 2 * q^73 - 5 * q^75 + 2 * q^81 - 5 * q^83 - 2 * q^89 - 2 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more