# Properties

 Label 1600.607 Modulus $1600$ Conductor $40$ Order $4$ Real no Primitive no Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(1600, base_ring=CyclotomicField(4))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(607,1600))

## Basic properties

 Modulus: $$1600$$ Conductor: $$40$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{40}(27,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1600.o

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: 4.4.8000.1

## Values on generators

$$(1151,901,577)$$ → $$(-1,-1,i)$$

## Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$i$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1600 }(607,a) \;$$ at $$\;a =$$ e.g. 2