# Properties

 Label 192.47 Modulus $192$ Conductor $48$ Order $4$ Real no Primitive no Minimal no Parity even

# Related objects

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

H = DirichletGroup(192, base_ring=CyclotomicField(4))

M = H._module

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

pari: [g,chi] = znchar(Mod(47,192))

## Basic properties

 Modulus: $$192$$ Conductor: $$48$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{48}(35,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 192.k

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\sqrt{-1})$$ Fixed field: 4.4.18432.1

## Values on generators

$$(127,133,65)$$ → $$(-1,-i,-1)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$\chi_{ 192 }(47, a)$$ $$1$$ $$1$$ $$i$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 192 }(47,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 192 }(47,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 192 }(47,·),\chi_{ 192 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 192 }(47,·)) \;$$ at $$\; a,b =$$ e.g. 1,2