LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(4))

M = H._module

chi = DirichletCharacter(H, M([0,1,0,0]))

pari: [g,chi] = znchar(Mod(181,720))

Basic properties

Modulus:	$720$
Conductor:	$16$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$4$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	no
Primitive:	no, induced from $\chi_{16}(5,\cdot)$	sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1
Minimal:	yes
Parity:	even	sage: chi.is_odd() pari: zncharisodd(g,chi)

Galois orbit 720.t

$\chi_{720}(181,\cdot)$ $\chi_{720}(541,\cdot)$

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:	$\Q(\zeta_{16})^+$

Values on generators

$(271,181,641,577)$ → $(1,i,1,1)$

First values

$a$	$-1$	$1$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$
$\chi_{ 720 }(181, a)$	$1$	$1$	$-1$	$i$	$-i$	$1$	$-i$	$-1$	$-i$	$1$	$i$	$-1$

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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