# Properties

 Label 512.g Modulus $512$ Conductor $32$ Order $8$ Real no Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(512, base_ring=CyclotomicField(8))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,7]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(65,512))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$512$$ Conductor: $$32$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 32.g sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $$\Q(\zeta_{8})$$ Fixed field: $$\Q(\zeta_{32})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{512}(65,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{512}(193,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{512}(321,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{512}(449,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$