# Properties

 Label 111.47 Modulus $111$ Conductor $111$ Order $6$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(111)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3,4]))

pari: [g,chi] = znchar(Mod(47,111))

## Basic properties

 Modulus: $$111$$ Conductor: $$111$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$6$$ 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 111.i

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Values on generators

$$(38,76)$$ → $$(-1,e\left(\frac{2}{3}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\sqrt{-3})$$ Fixed field: 6.0.50602347.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 111 }(47,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{111}(47,\cdot)) = \sum_{r\in \Z/111\Z} \chi_{111}(47,r) e\left(\frac{2r}{111}\right) = -8.2534858849+-6.5482799839i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 111 }(47,·),\chi_{ 111 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{111}(47,\cdot),\chi_{111}(1,\cdot)) = \sum_{r\in \Z/111\Z} \chi_{111}(47,r) \chi_{111}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 111 }(47,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{111}(47,·)) = \sum_{r \in \Z/111\Z} \chi_{111}(47,r) e\left(\frac{1 r + 2 r^{-1}}{111}\right) = -0.0$$