# Properties

 Label 20.e Modulus $20$ Conductor $20$ Order $4$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(20, base_ring=CyclotomicField(4))

M = H._module

chi = DirichletCharacter(H, M([2,3]))

chi.galois_orbit()

[g,chi] = znchar(Mod(3,20))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$20$$ Conductor: $$20$$ 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: $$\Q(\zeta_{20})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$
$$\chi_{20}(3,\cdot)$$ $$1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$
$$\chi_{20}(7,\cdot)$$ $$1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$i$$