# Properties

 Label 1045.417 Modulus $1045$ Conductor $1045$ Order $4$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(1045, base_ring=CyclotomicField(4))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,2,2]))

pari: [g,chi] = znchar(Mod(417,1045))

## Basic properties

 Modulus: $$1045$$ Conductor: $$1045$$ 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: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1045.k

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: 4.0.5460125.4

## Values on generators

$$(837,761,496)$$ → $$(i,-1,-1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$ $$\chi_{ 1045 }(417, a)$$ $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1045 }(417,a) \;$$ at $$\;a =$$ e.g. 2