# Properties

 Conductor 231 Order 30 Real no Primitive yes Minimal yes Parity even Orbit label 231.bc

# 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(231)

sage: chi = H[5]

pari: [g,chi] = znchar(Mod(5,231))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 231 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 30 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 = 231.bc Orbit index = 29

## 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

$$(155,199,211)$$ → $$(-1,e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right))$$

## Values

 -1 1 2 4 5 8 10 13 16 17 19 20 $$1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{15})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 231 }(5,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{231}(5,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(5,r) e\left(\frac{2r}{231}\right) = -14.2715465691+-5.2271367427i$$

## Jacobi sum

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

$$J(\chi_{ 231 }(5,·),\chi_{ 231 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{231}(5,\cdot),\chi_{231}(1,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(5,r) \chi_{231}(1,1-r) = -1$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 231 }(5,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{231}(5,·)) = \sum_{r \in \Z/231\Z} \chi_{231}(5,r) e\left(\frac{1 r + 2 r^{-1}}{231}\right) = -0.0$$