# Properties

 Conductor 27 Order 18 Real No Primitive No Parity Odd Orbit Label 81.f

# Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(81)
sage: chi = H[17]
pari: [g,chi] = znchar(Mod(17,81))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 27 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 18 Real = No sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = No sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 81.f Orbit index = 6

## Galois orbit

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

## Values on generators

$$2$$ → $$e\left(\frac{11}{18}\right)$$

## Values

 -1 1 2 4 5 7 8 10 11 13 14 16 $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{9})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 81 }(17,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{81}(17,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(17,r) e\left(\frac{2r}{81}\right) = -0.0$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 81 }(17,·),\chi_{ 81 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{81}(17,\cdot),\chi_{81}(1,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(17,r) \chi_{81}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 81 }(17,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{81}(17,·)) = \sum_{r \in \Z/81\Z} \chi_{81}(17,r) e\left(\frac{1 r + 2 r^{-1}}{81}\right) = -0.0$$