# Properties

 Conductor 40 Order 2 Real yes Primitive yes Minimal yes Parity odd Orbit label 40.e

# Related objects

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

sage: H = DirichletGroup_conrey(40)

sage: chi = H[19]

pari: [g,chi] = znchar(Mod(19,40))

## Kronecker symbol representation

sage: kronecker_character(-40)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{-40}{\bullet}\right)$$

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 40 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 2 Real = yes 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 = 40.e Orbit index = 5

## 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

$$(31,21,17)$$ → $$(-1,-1,-1)$$

## Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 40 }(19,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{40}(19,\cdot)) = \sum_{r\in \Z/40\Z} \chi_{40}(19,r) e\left(\frac{r}{20}\right) = 0.0$$

## Jacobi sum

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

$$J(\chi_{ 40 }(19,·),\chi_{ 40 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{40}(19,\cdot),\chi_{40}(1,\cdot)) = \sum_{r\in \Z/40\Z} \chi_{40}(19,r) \chi_{40}(1,1-r) = 0$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 40 }(19,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{40}(19,·)) = \sum_{r \in \Z/40\Z} \chi_{40}(19,r) e\left(\frac{1 r + 2 r^{-1}}{40}\right) = 0.0$$