# Properties

 Conductor 569 Order 71 Real No Primitive Yes Parity Even Orbit Label 569.e

# 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(569)
sage: chi = H[142]
pari: [g,chi] = znchar(Mod(142,569))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 569 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 71 Real = No sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Even Orbit label = 569.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

$$3$$ → $$e\left(\frac{50}{71}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{11}{71}\right)$$ $$e\left(\frac{50}{71}\right)$$ $$e\left(\frac{22}{71}\right)$$ $$e\left(\frac{32}{71}\right)$$ $$e\left(\frac{61}{71}\right)$$ $$e\left(\frac{24}{71}\right)$$ $$e\left(\frac{33}{71}\right)$$ $$e\left(\frac{29}{71}\right)$$ $$e\left(\frac{43}{71}\right)$$ $$e\left(\frac{45}{71}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{71})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 569 }(142,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{569}(142,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(142,r) e\left(\frac{2r}{569}\right) = 4.1045485781+-23.4979292911i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 569 }(142,·),\chi_{ 569 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{569}(142,\cdot),\chi_{569}(1,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(142,r) \chi_{569}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 569 }(142,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{569}(142,·)) = \sum_{r \in \Z/569\Z} \chi_{569}(142,r) e\left(\frac{1 r + 2 r^{-1}}{569}\right) = -1.5234139101+-0.8061746713i$$