# Properties

 Label 57.d Modulus $57$ Conductor $57$ Order $2$ Real yes Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(57, base_ring=CyclotomicField(2))

M = H._module

chi = DirichletCharacter(H, M([1,1]))

chi.galois_orbit()

[g,chi] = znchar(Mod(56,57))

order = charorder(g,chi)

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

## Kronecker symbol representation

sage: kronecker_character(57)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{57}{\bullet}\right)$$

## Basic properties

 Modulus: $$57$$ Conductor: $$57$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes 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: $$\Q$$ Fixed field: $$\Q(\sqrt{57})$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{57}(56,\cdot)$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$