# Properties

 Conductor 2003 Order 22 Real No Primitive Yes Parity Odd Orbit Label 2003.g

# 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(2003)

sage: chi = H[180]

pari: [g,chi] = znchar(Mod(180,2003))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 2003 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 22 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 = Odd Orbit label = 2003.g Orbit index = 7

## 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

$$5$$ → $$e\left(\frac{15}{22}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$-1$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{11})$$