# Properties

 Label 448.239 Modulus $448$ Conductor $16$ Order $4$ Real no Primitive no Minimal no Parity odd

# Learn more

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

sage: H = DirichletGroup(448, base_ring=CyclotomicField(4))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([2,3,0]))

pari: [g,chi] = znchar(Mod(239,448))

## Basic properties

 Modulus: $$448$$ 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 $$\chi_{16}(3,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 448.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: 4.0.2048.2

## Values on generators

$$(127,197,129)$$ → $$(-1,-i,1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$\chi_{ 448 }(239, a)$$ $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$1$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 448 }(239,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 448 }(239,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 448 }(239,·),\chi_{ 448 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 448 }(239,·)) \;$$ at $$\; a,b =$$ e.g. 1,2