LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(2))

M = H._module

chi = DirichletCharacter(H, M([1,1]))

pari: [g,chi] = znchar(Mod(208,209))

Kronecker symbol representation

sage: kronecker_character(209)

pari: znchartokronecker(g,chi)

$\displaystyle\left(\frac{209}{\bullet}\right)$

Basic properties

Modulus:	$209$
Conductor:	$209$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$2$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	yes
Primitive:	yes	sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1
Minimal:	yes
Parity:	even	sage: chi.is_odd() pari: zncharisodd(g,chi)

Galois orbit 209.d

$\chi_{209}(208,\cdot)$

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:	$\Q(\sqrt{209})$

Values on generators

$(134,78)$ → $(-1,-1)$

First values

$a$	$-1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$12$
$\chi_{ 209 }(208, a)$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$

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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