# Properties

 Conductor 10003 Order 3 Real No Primitive Yes Parity Even

# 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(10003)
sage: chi = H[2193]
pari: [g,chi] = znchar(Mod(2193,10003))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 10003 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 3 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

## 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

sage: chi(k) for k in H.gens()
pari: [ chareval(g,chi,x) | x <- g.gen ] \\ value in Q/Z

$$(2864,2859)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right))$$

## First values

 1 2 3 4 5 6 8 9 10 11 12 13 15 16 17 18 19 20 22 23 24 25 26 27 29 30 31 32 33 34 1 1 $$e\left(\frac{2}{3}\right)$$ 1 $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ 1 $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ 1 1 $$e\left(\frac{1}{3}\right)$$ 1 $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ 1 $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ 1 $$e\left(\frac{1}{3}\right)$$ 1
value at  e.g. 2

## Related number fields

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