# Properties

 Conductor 17 Order 16 Real no Primitive no Minimal yes Parity odd Orbit label 85.q

# Related objects

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(85)

sage: chi = H[6]

pari: [g,chi] = znchar(Mod(6,85))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 17 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 16 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 85.q Orbit index = 17

## 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

$$(52,71)$$ → $$(1,e\left(\frac{15}{16}\right))$$

## Values

 -1 1 2 3 4 6 7 8 9 11 12 13 $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{16})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 85 }(6,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{85}(6,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(6,r) e\left(\frac{2r}{85}\right) = -3.4512252809+2.2558909682i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)

$$J(\chi_{ 85 }(6,·),\chi_{ 85 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{85}(6,\cdot),\chi_{85}(1,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(6,r) \chi_{85}(1,1-r) = -3$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)

$$K(a,b,\chi_{ 85 }(6,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{85}(6,·)) = \sum_{r \in \Z/85\Z} \chi_{85}(6,r) e\left(\frac{1 r + 2 r^{-1}}{85}\right) = -3.1743261592+7.6635012649i$$