Properties

 Label 4560.1709 Modulus $4560$ Conductor $4560$ Order $4$ Real no Primitive yes Minimal yes Parity even

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

sage: H = DirichletGroup(4560, base_ring=CyclotomicField(4))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3,2,2,2]))

pari: [g,chi] = znchar(Mod(1709,4560))

Basic properties

 Modulus: $$4560$$ Conductor: $$4560$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 4560.cz

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Related number fields

 Field of values: $$\Q(\sqrt{-1})$$ Fixed field: 4.4.166348800.1

Values on generators

$$(1711,1141,3041,2737,1921)$$ → $$(1,-i,-1,-1,-1)$$

Values

 $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$1$$ $$i$$ $$i$$ $$1$$ $$-1$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$i$$
sage: chi.jacobi_sum(n)

$$\chi_{ 4560 }(1709,a) \;$$ at $$\;a =$$ e.g. 2