LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

M = H._module

chi = DirichletCharacter(H, M([0,0]))

pari: [g,chi] = znchar(Mod(1,16))

Basic properties

Modulus:	$16$
Conductor:	$1$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$1$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	yes
Primitive:	no, induced from $\chi_{1}(0,\cdot)$	sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1
Minimal:	no
Parity:	even	sage: chi.is_odd() pari: zncharisodd(g,chi)

Galois orbit 16.a

$\chi_{16}(1,\cdot)$

sage: chi.galois_orbit()

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Related number fields

Field of values:	$\Q$
Fixed field:	$\Q$

Values on generators

$(15,5)$ → $(1,1)$

Values

$a$	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$
$\chi_{ 16 }(1, a)$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$

sage: chi.jacobi_sum(n)

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

Jacobi sum

sage: chi.jacobi_sum(n)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

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