LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(4))

M = H._module

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

pari: [g,chi] = znchar(Mod(3,20))

Basic properties

Modulus:	$20$
Conductor:	$20$	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 20.e

$\chi_{20}(3,\cdot)$ $\chi_{20}(7,\cdot)$

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:	$\Q(\zeta_{20})^+$

Values on generators

$(11,17)$ → $(-1,-i)$

Values

$a$	$-1$	$1$	$3$	$7$	$9$	$11$	$13$	$17$
$\chi_{ 20 }(3, a)$	$1$	$1$	$-i$	$i$	$-1$	$-1$	$i$	$-i$

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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