# Properties

 Modulus 4000 Conductor 125 Order 100 Real no Primitive no Minimal yes Parity odd Orbit label 4000.cq

# Learn more about

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

sage: H = DirichletGroup(4000)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(33,4000))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 4000 Conductor = 125 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 100 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 4000.cq Orbit index = 69

## 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

$$(2751,2501,1377)$$ → $$(1,1,e\left(\frac{83}{100}\right))$$

## Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$-1$$ $$1$$ $$e\left(\frac{81}{100}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{37}{100}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{43}{100}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{100})$$