# Properties

 Conductor 36 Order 6 Real no Primitive yes Minimal yes Parity odd Orbit label 36.f

# Learn more about

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

sage: H = DirichletGroup_conrey(36)

sage: chi = H[31]

pari: [g,chi] = znchar(Mod(31,36))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 36 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 = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 36.f Orbit index = 6

## 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

$$(19,29)$$ → $$(-1,e\left(\frac{1}{3}\right))$$

## Values

 -1 1 5 7 11 13 17 19 23 25 29 31 $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
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)

$$\tau_{ a }( \chi_{ 36 }(31,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{36}(31,\cdot)) = \sum_{r\in \Z/36\Z} \chi_{36}(31,r) e\left(\frac{r}{18}\right) = -0.0$$

## Jacobi sum

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

$$J(\chi_{ 36 }(31,·),\chi_{ 36 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{36}(31,\cdot),\chi_{36}(1,\cdot)) = \sum_{r\in \Z/36\Z} \chi_{36}(31,r) \chi_{36}(1,1-r) = 0$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 36 }(31,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{36}(31,·)) = \sum_{r \in \Z/36\Z} \chi_{36}(31,r) e\left(\frac{1 r + 2 r^{-1}}{36}\right) = -5.1961524227+3.0i$$