# Properties

 Conductor 91 Order 12 Real No Primitive Yes Parity Odd Orbit Label 91.z

# 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(91)
sage: chi = H[18]
pari: [g,chi] = znchar(Mod(18,91))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 91 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 12 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 = 91.z Orbit index = 26

## 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

$$(66,15)$$ → $$(e\left(\frac{2}{3}\right),-i)$$

## Values

 -1 1 2 3 4 5 6 8 9 10 11 12 $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{12})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 91 }(18,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{91}(18,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(18,r) e\left(\frac{2r}{91}\right) = -2.1805818798+9.2868219895i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 91 }(18,·),\chi_{ 91 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{91}(18,\cdot),\chi_{91}(1,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(18,r) \chi_{91}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 91 }(18,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{91}(18,·)) = \sum_{r \in \Z/91\Z} \chi_{91}(18,r) e\left(\frac{1 r + 2 r^{-1}}{91}\right) = 4.8367535439+-18.0510099697i$$