# Properties

 Modulus 4008 Conductor 4008 Order 166 Real no Primitive yes Minimal yes Parity odd Orbit label 4008.r

# Related objects

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

sage: H = DirichletGroup(4008)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,83,83,146]))

pari: [g,chi] = znchar(Mod(77,4008))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 4008 Conductor = 4008 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 166 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 4008.r Orbit index = 18

## Galois orbit

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Values on generators

$$(3007,2005,1337,673)$$ → $$(1,-1,-1,e\left(\frac{73}{83}\right))$$

## Values

 -1 1 5 7 11 13 17 19 23 25 29 31 $$-1$$ $$1$$ $$e\left(\frac{73}{83}\right)$$ $$e\left(\frac{65}{83}\right)$$ $$e\left(\frac{52}{83}\right)$$ $$e\left(\frac{15}{166}\right)$$ $$e\left(\frac{19}{166}\right)$$ $$e\left(\frac{85}{166}\right)$$ $$e\left(\frac{95}{166}\right)$$ $$e\left(\frac{63}{83}\right)$$ $$e\left(\frac{77}{83}\right)$$ $$e\left(\frac{13}{83}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{83})$$