# Properties

 Label 185.108 Modulus $185$ Conductor $185$ Order $36$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(185, base_ring=CyclotomicField(36))

M = H._module

chi = DirichletCharacter(H, M([27,8]))

pari: [g,chi] = znchar(Mod(108,185))

## Basic properties

 Modulus: $$185$$ Conductor: $$185$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$36$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no 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 185.bd

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Values on generators

$$(112,76)$$ → $$(-i,e\left(\frac{2}{9}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$\chi_{ 185 }(108, a)$$ $$-1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 185 }(108,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 185 }(108,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 185 }(108,·),\chi_{ 185 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 185 }(108,·)) \;$$ at $$\; a,b =$$ e.g. 1,2