# Properties

 Label 111.68 Modulus $111$ Conductor $111$ Order $4$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(111, base_ring=CyclotomicField(4))

M = H._module

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

pari: [g,chi] = znchar(Mod(68,111))

## Basic properties

 Modulus: $$111$$ Conductor: $$111$$ 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: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 111.g

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: 4.4.455877.1

## Values on generators

$$(38,76)$$ → $$(-1,i)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$\chi_{ 111 }(68, a)$$ $$1$$ $$1$$ $$-i$$ $$-1$$ $$i$$ $$1$$ $$i$$ $$1$$ $$1$$ $$-i$$ $$-i$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 111 }(68,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 111 }(68,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 111 }(68,·),\chi_{ 111 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 111 }(68,·)) \;$$ at $$\; a,b =$$ e.g. 1,2