# Properties

 Label 119.111 Modulus $119$ Conductor $119$ Order $8$ 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(119, base_ring=CyclotomicField(8))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([4,1]))

pari: [g,chi] = znchar(Mod(111,119))

## Basic properties

 Modulus: $$119$$ Conductor: $$119$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ 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 119.l

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: $$\Q(\zeta_{8})$$ Fixed field: 8.0.985223153873.1

## Values on generators

$$(52,71)$$ → $$(-1,e\left(\frac{1}{8}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 119 }(111,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{119}(111,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(111,r) e\left(\frac{2r}{119}\right) = -0.5074023908+10.8969051943i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 119 }(111,·),\chi_{ 119 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{119}(111,\cdot),\chi_{119}(1,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(111,r) \chi_{119}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 119 }(111,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{119}(111,·)) = \sum_{r \in \Z/119\Z} \chi_{119}(111,r) e\left(\frac{1 r + 2 r^{-1}}{119}\right) = -2.6114006516+-2.6114006516i$$