# Properties

 Label 65.f Modulus $65$ Conductor $65$ Order $4$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(65, base_ring=CyclotomicField(4))

M = H._module

chi = DirichletCharacter(H, M([3,3]))

chi.galois_orbit()

[g,chi] = znchar(Mod(18,65))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$65$$ Conductor: $$65$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ 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)

## Related number fields

 Field of values: $$\mathbb{Q}(i)$$ Fixed field: 4.4.274625.2

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$14$$
$$\chi_{65}(18,\cdot)$$ $$1$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$-i$$ $$1$$ $$-1$$ $$-1$$ $$i$$ $$i$$ $$-1$$
$$\chi_{65}(47,\cdot)$$ $$1$$ $$1$$ $$-1$$ $$-i$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$