# Properties

 Label 6003.20 Modulus $6003$ Conductor $6003$ Order $462$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(6003)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([77,105,396]))

pari: [g,chi] = znchar(Mod(20,6003))

## Basic properties

 Modulus: $$6003$$ Conductor: $$6003$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$462$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 6003.dl

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

$$(668,3133,4555)$$ → $$(e\left(\frac{1}{6}\right),e\left(\frac{5}{22}\right),e\left(\frac{6}{7}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{221}{462}\right)$$ $$e\left(\frac{221}{231}\right)$$ $$e\left(\frac{212}{231}\right)$$ $$e\left(\frac{125}{462}\right)$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{148}{231}\right)$$ $$e\left(\frac{218}{231}\right)$$ $$e\left(\frac{173}{231}\right)$$ $$e\left(\frac{211}{231}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{231})$ Fixed field: Number field defined by a degree 462 polynomial