# Properties

 Conductor 32 Order 8 Real No Primitive No Parity Even Orbit Label 96.n

# Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(96)
sage: chi = H[85]
pari: [g,chi] = znchar(Mod(85,96))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 32 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 8 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 = Even Orbit label = 96.n Orbit index = 14

## 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

$$(31,37,65)$$ → $$(1,e\left(\frac{5}{8}\right),1)$$

## Values

 -1 1 5 7 11 13 17 19 23 25 29 31 $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{8})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 96 }(85,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{96}(85,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(85,r) e\left(\frac{r}{48}\right) = 0.0$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 96 }(85,·),\chi_{ 96 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{96}(85,\cdot),\chi_{96}(1,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(85,r) \chi_{96}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 96 }(85,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{96}(85,·)) = \sum_{r \in \Z/96\Z} \chi_{96}(85,r) e\left(\frac{1 r + 2 r^{-1}}{96}\right) = 6.2855596671+9.4070048194i$$