# Properties

 Conductor 291 Order 32 Real No Primitive Yes Parity Even Orbit Label 291.s

# 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(291)

sage: chi = H[116]

pari: [g,chi] = znchar(Mod(116,291))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 291 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 32 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 = 291.s Orbit index = 19

## 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

$$(98,199)$$ → $$(-1,e\left(\frac{27}{32}\right))$$

## Values

 -1 1 2 4 5 7 8 10 11 13 14 16 $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$-i$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{32})$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 291 }(116,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{291}(116,\cdot)) = \sum_{r\in \Z/291\Z} \chi_{291}(116,r) e\left(\frac{2r}{291}\right) = -4.8925715005+16.3420544642i$$

## Jacobi sum

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

$$J(\chi_{ 291 }(116,·),\chi_{ 291 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{291}(116,\cdot),\chi_{291}(1,\cdot)) = \sum_{r\in \Z/291\Z} \chi_{291}(116,r) \chi_{291}(1,1-r) = 1$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 291 }(116,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{291}(116,·)) = \sum_{r \in \Z/291\Z} \chi_{291}(116,r) e\left(\frac{1 r + 2 r^{-1}}{291}\right) = -0.0$$