# Properties

 Conductor 85 Order 4 Real No Primitive Yes Parity Odd Orbit Label 85.i

# 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(85)
sage: chi = H[38]
pari: [g,chi] = znchar(Mod(38,85))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 85 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 4 Real = No sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 85.i Orbit index = 9

## 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

$$(52,71)$$ → $$(-i,-i)$$

## Values

 -1 1 2 3 4 6 7 8 9 11 12 13 $$-1$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$-i$$ $$1$$ $$i$$ $$-1$$ $$i$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(i)$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 85 }(38,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{85}(38,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(38,r) e\left(\frac{2r}{85}\right) = -1.0060069295+9.1644939881i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 85 }(38,·),\chi_{ 85 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{85}(38,\cdot),\chi_{85}(1,\cdot)) = \sum_{r\in \Z/85\Z} \chi_{85}(38,r) \chi_{85}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 85 }(38,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{85}(38,·)) = \sum_{r \in \Z/85\Z} \chi_{85}(38,r) e\left(\frac{1 r + 2 r^{-1}}{85}\right) = 3.9601822827+-3.9601822827i$$