# Properties

 Conductor 10009 Order 3 Real No Primitive Yes Parity Even

# Related objects

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

sage: H = DirichletGroup_conrey(10009)

sage: chi = H[8964]

pari: [g,chi] = znchar(Mod(8964,10009))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 10009 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

$$11$$ → $$e\left(\frac{1}{3}\right)$$

## First values

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 1 $$e\left(\frac{1}{3}\right)$$ 1 $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ 1 1 $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ 1 1 1 1 $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ 1 $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ 1 1 1 $$e\left(\frac{1}{3}\right)$$ 1
value at  e.g. 2

## Related number fields

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