# Properties

 Label 6003.4187 Modulus $6003$ Conductor $261$ Order $84$ Real no Primitive no Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(6003, base_ring=CyclotomicField(84))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([14,0,75]))

pari: [g,chi] = znchar(Mod(4187,6003))

## Basic properties

 Modulus: $$6003$$ Conductor: $$261$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$84$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{261}(11,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 6003.cq

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

$$(668,3133,4555)$$ → $$(e\left(\frac{1}{6}\right),1,e\left(\frac{25}{28}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{5}{21}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{84})$ Fixed field: Number field defined by a degree 84 polynomial