# Properties

 Conductor 21 Order 6 Real No Primitive No Parity Odd Orbit Label 84.p

# 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(84)
sage: chi = H[65]
pari: [g,chi] = znchar(Mod(65,84))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 21 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 6 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 = 84.p Orbit index = 16

## 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

$$(43,29,73)$$ → $$(1,-1,e\left(\frac{1}{3}\right))$$

## Values

 -1 1 5 11 13 17 19 23 25 29 31 37 $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{3})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 84 }(65,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{84}(65,\cdot)) = \sum_{r\in \Z/84\Z} \chi_{84}(65,r) e\left(\frac{r}{42}\right) = 4.0706876037+8.2115468965i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 84 }(65,·),\chi_{ 84 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{84}(65,\cdot),\chi_{84}(1,\cdot)) = \sum_{r\in \Z/84\Z} \chi_{84}(65,r) \chi_{84}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 84 }(65,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{84}(65,·)) = \sum_{r \in \Z/84\Z} \chi_{84}(65,r) e\left(\frac{1 r + 2 r^{-1}}{84}\right) = 0.0$$