# Properties

 Label 164.163 Modulus $164$ Conductor $164$ Order $2$ Real yes Primitive yes Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(164, base_ring=CyclotomicField(2))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(163,164))

## Kronecker symbol representation

sage: kronecker_character(-164)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{-164}{\bullet}\right)$$

## Basic properties

 Modulus: $$164$$ Conductor: $$164$$ 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: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 164.d

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: $$\Q$$ Fixed field: $$\Q(\sqrt{-41})$$

## Values on generators

$$(83,129)$$ → $$(-1,-1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$\chi_{ 164 }(163, a)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 164 }(163,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 164 }(163,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 164 }(163,·),\chi_{ 164 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 164 }(163,·)) \;$$ at $$\; a,b =$$ e.g. 1,2