Properties

 Label 5184.4535 Modulus $5184$ Conductor $96$ Order $8$ Real no Primitive no Minimal no Parity even

Related objects

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

H = DirichletGroup(5184, base_ring=CyclotomicField(8))

M = H._module

chi = DirichletCharacter(H, M([4,3,4]))

pari: [g,chi] = znchar(Mod(4535,5184))

Basic properties

 Modulus: $$5184$$ Conductor: $$96$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{96}(35,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 5184.w

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: $$\Q(\zeta_{8})$$ Fixed field: 8.8.173946175488.1

Values on generators

$$(2431,325,1217)$$ → $$(-1,e\left(\frac{3}{8}\right),-1)$$

First values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$\chi_{ 5184 }(4535, a)$$ $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 5184 }(4535,a) \;$$ at $$\;a =$$ e.g. 2