LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(58, base_ring=CyclotomicField(4))

M = H._module

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

pari: [g,chi] = znchar(Mod(41,58))

Basic properties

Modulus:	$58$
Conductor:	$29$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$4$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	no
Primitive:	no, induced from $\chi_{29}(12,\cdot)$	sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1
Minimal:	yes
Parity:	odd	sage: chi.is_odd() pari: zncharisodd(g,chi)

Galois orbit 58.c

$\chi_{58}(17,\cdot)$ $\chi_{58}(41,\cdot)$

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:	4.0.24389.1

Values on generators

$31$ → $i$

Values

$a$	$-1$	$1$	$3$	$5$	$7$	$9$	$11$	$13$	$15$	$17$	$19$	$21$
$\chi_{ 58 }(41, a)$	$-1$	$1$	$i$	$-1$	$1$	$-1$	$i$	$-1$	$-i$	$i$	$i$	$i$

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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