# Properties

 Label 162.89 Modulus $162$ Conductor $27$ Order $18$ Real no Primitive no Minimal no Parity odd

# Learn more

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

H = DirichletGroup(162, base_ring=CyclotomicField(18))

M = H._module

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

pari: [g,chi] = znchar(Mod(89,162))

## Basic properties

 Modulus: $$162$$ Conductor: $$27$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$18$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{27}(2,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 162.f

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: $$\Q(\zeta_{9})$$ Fixed field: Number field defined by a degree 18 polynomial

## Values on generators

$$83$$ → $$e\left(\frac{1}{18}\right)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$\chi_{ 162 }(89, a)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 162 }(89,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 162 }(89,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 162 }(89,·),\chi_{ 162 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 162 }(89,·)) \;$$ at $$\; a,b =$$ e.g. 1,2