# Properties

 Conductor 6003 Order 6 Real No Primitive Yes Parity Even Orbit Label 6003.n

# Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(6003)

sage: chi = H[2000]

pari: [g,chi] = znchar(Mod(2000,6003))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 6003 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 6 Real = No sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = Even Orbit label = 6003.n Orbit index = 14

## Galois orbit

sage: chi.sage_character().galois_orbit()

pari: order = charorder(g,chi)

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

## Values on generators

$$(668,3133,4555)$$ → $$(e\left(\frac{1}{6}\right),-1,-1)$$

## Values

 -1 1 2 4 5 7 8 10 11 13 14 16 $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{3})$$