# Properties

 Label 967.239 Modulus $967$ Conductor $967$ Order $138$ Real no Primitive yes Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(967, base_ring=CyclotomicField(138))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([89]))

pari: [g,chi] = znchar(Mod(239,967))

## Basic properties

 Modulus: $$967$$ Conductor: $$967$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$138$$ 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 967.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: $\Q(\zeta_{69})$ Fixed field: Number field defined by a degree 138 polynomial (not computed)

## Values on generators

$$5$$ → $$e\left(\frac{89}{138}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$-1$$ $$1$$ $$e\left(\frac{14}{69}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{28}{69}\right)$$ $$e\left(\frac{89}{138}\right)$$ $$e\left(\frac{7}{138}\right)$$ $$e\left(\frac{29}{138}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{16}{23}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 967 }(239,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 967 }(239,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 967 }(239,·),\chi_{ 967 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 967 }(239,·)) \;$$ at $$\; a,b =$$ e.g. 1,2