# Properties

 Label 1323.667 Modulus $1323$ Conductor $63$ Order $3$ Real no Primitive no Minimal no Parity even

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(6))

M = H._module

chi = DirichletCharacter(H, M([4,2]))

pari: [g,chi] = znchar(Mod(667,1323))

## Basic properties

 Modulus: $$1323$$ Conductor: $$63$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$3$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{63}(16,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1323.g

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\sqrt{-3})$$ Fixed field: 3.3.3969.2

## Values on generators

$$(785,1081)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 1323 }(667, a)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1323 }(667,a) \;$$ at $$\;a =$$ e.g. 2