Properties

 Label 4560.239 Modulus $4560$ Conductor $1140$ Order $6$ Real no Primitive no Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(4560, base_ring=CyclotomicField(6))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3,0,3,3,4]))

pari: [g,chi] = znchar(Mod(239,4560))

Basic properties

 Modulus: $$4560$$ Conductor: $$1140$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$6$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{1140}(239,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 4560.ef

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: $$\Q(\sqrt{-3})$$ Fixed field: 6.6.28149336000.1

Values on generators

$$(1711,1141,3041,2737,1921)$$ → $$(-1,1,-1,-1,e\left(\frac{2}{3}\right))$$

Values

 $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 4560 }(239,a) \;$$ at $$\;a =$$ e.g. 2