# Properties

 Label 193.192 Modulus $193$ Conductor $193$ Order $2$ Real yes Primitive yes Minimal yes Parity even

# Learn more

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

H = DirichletGroup(193, base_ring=CyclotomicField(2))

M = H._module

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

pari: [g,chi] = znchar(Mod(192,193))

## Kronecker symbol representation

sage: kronecker_character(193)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{193}{\bullet}\right)$$

## Basic properties

 Modulus: $$193$$ Conductor: $$193$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes 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 193.b

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: $$\Q$$ Fixed field: $$\Q(\sqrt{193})$$

## Values on generators

$$5$$ → $$-1$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$\chi_{ 193 }(192, a)$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 193 }(192,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 193 }(192,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 193 }(192,·),\chi_{ 193 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 193 }(192,·)) \;$$ at $$\; a,b =$$ e.g. 1,2