Properties

 Label 8.7 Modulus $8$ Conductor $4$ Order $2$ Real yes Primitive no Minimal no Parity odd

Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

M = H._module

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

pari: [g,chi] = znchar(Mod(7,8))

Basic properties

 Modulus: $$8$$ Conductor: $$4$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{4}(3,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 8.c

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: $$\Q$$ Fixed field: $$\Q(\sqrt{-1})$$

Values on generators

$$(7,5)$$ → $$(-1,1)$$

Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$\chi_{ 8 }(7, a)$$ $$-1$$ $$1$$ $$-1$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 8 }(7,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 8 }(7,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 8 }(7,·),\chi_{ 8 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 8 }(7,·)) \;$$ at $$\; a,b =$$ e.g. 1,2