# Properties

 Label 1792.1345 Modulus $1792$ Conductor $16$ Order $4$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(1792, base_ring=CyclotomicField(4))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3,0]))

pari: [g,chi] = znchar(Mod(1345,1792))

## Basic properties

 Modulus: $$1792$$ Conductor: $$16$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{16}(13,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1792.m

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\sqrt{-1})$$ Fixed field: $$\Q(\zeta_{16})^+$$

## Values on generators

$$(1023,1541,1025)$$ → $$(1,-i,1)$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-1$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1792 }(1345,a) \;$$ at $$\;a =$$ e.g. 2