LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(680, base_ring=CyclotomicField(4))

M = H._module

chi = DirichletCharacter(H, M([2,2,3,3]))

pari: [g,chi] = znchar(Mod(123,680))

Basic properties

Modulus:	$680$
Conductor:	$680$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$4$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	no
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 680.bl

$\chi_{680}(123,\cdot)$ $\chi_{680}(387,\cdot)$

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:	$\mathbb{Q}(i)$
Fixed field:	4.4.39304000.1

Values on generators

$(511,341,137,241)$ → $(-1,-1,-i,-i)$

First values

$a$	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$19$	$21$	$23$	$27$	$29$
$\chi_{ 680 }(123, a)$	$1$	$1$	$1$	$-1$	$1$	$i$	$-i$	$1$	$-1$	$1$	$1$	$-i$

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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