LMFDB - Dirichlet character \(\chi

Show commands: PariGP / SageMath

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(728, base_ring=CyclotomicField(6))

M = H._module

chi = DirichletCharacter(H, M([0,0,2,4]))

pari: [g,chi] = znchar(Mod(9,728))

Basic properties

Modulus:	$728$
Conductor:	$91$	sage: chi.conductor() pari: znconreyconductor(g,chi)
Order:	$3$	sage: chi.multiplicative_order() pari: charorder(g,chi)
Real:	no
Primitive:	no, induced from $\chi_{91}(9,\cdot)$	sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1
Minimal:	yes
Parity:	even	sage: chi.is_odd() pari: zncharisodd(g,chi)

Galois orbit 728.t

$\chi_{728}(9,\cdot)$ $\chi_{728}(81,\cdot)$

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:	$\mathbb{Q}(\zeta_3)$
Fixed field:	3.3.8281.2

Values on generators

$(183,365,521,561)$ → $(1,1,e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right))$

First values

$a$	$-1$	$1$	$3$	$5$	$9$	$11$	$15$	$17$	$19$	$23$	$25$	$27$
$\chi_{ 728 }(9, a)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$	$1$

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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