# Properties

 Label 6003.82 Modulus $6003$ Conductor $667$ Order $77$ Real no Primitive no 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([0,49,22]))

pari: [g,chi] = znchar(Mod(82,6003))

## Basic properties

 Modulus: $$6003$$ Conductor: $$667$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$77$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{667}(82,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 6003.cm

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

$$(668,3133,4555)$$ → $$(1,e\left(\frac{7}{11}\right),e\left(\frac{2}{7}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{43}{77}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{71}{77}\right)$$ $$e\left(\frac{40}{77}\right)$$ $$e\left(\frac{52}{77}\right)$$ $$e\left(\frac{37}{77}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{4}{77}\right)$$ $$e\left(\frac{6}{77}\right)$$ $$e\left(\frac{18}{77}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{77})$ Fixed field: Number field defined by a degree 77 polynomial