# Properties

 Conductor 504 Order 6 Real no Primitive yes Minimal yes Parity even Orbit label 504.cy

# 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(504)

sage: chi = H[347]

pari: [g,chi] = znchar(Mod(347,504))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 504 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 = even Orbit label = 504.cy Orbit index = 77

## 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

$$(127,253,281,73)$$ → $$(-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right))$$

## Values

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

$$\tau_{ a }( \chi_{ 504 }(347,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{504}(347,\cdot)) = \sum_{r\in \Z/504\Z} \chi_{504}(347,r) e\left(\frac{r}{252}\right) = 0.0$$

## Jacobi sum

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

$$J(\chi_{ 504 }(347,·),\chi_{ 504 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{504}(347,\cdot),\chi_{504}(1,\cdot)) = \sum_{r\in \Z/504\Z} \chi_{504}(347,r) \chi_{504}(1,1-r) = 0$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 504 }(347,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{504}(347,·)) = \sum_{r \in \Z/504\Z} \chi_{504}(347,r) e\left(\frac{1 r + 2 r^{-1}}{504}\right) = -9.9711075313+-5.7568216173i$$