# Properties

 Conductor 88 Order 10 Real No Primitive Yes Parity Odd Orbit Label 88.l

# 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(88)
sage: chi = H[27]
pari: [g,chi] = znchar(Mod(27,88))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 88 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 10 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 = Odd Orbit label = 88.l Orbit index = 12

## 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

$$(23,45,57)$$ → $$(-1,-1,e\left(\frac{2}{5}\right))$$

## Values

 -1 1 3 5 7 9 13 15 17 19 21 23 $$-1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$-1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{5})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 88 }(27,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{88}(27,\cdot)) = \sum_{r\in \Z/88\Z} \chi_{88}(27,r) e\left(\frac{r}{44}\right) = 0.0$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 88 }(27,·),\chi_{ 88 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{88}(27,\cdot),\chi_{88}(1,\cdot)) = \sum_{r\in \Z/88\Z} \chi_{88}(27,r) \chi_{88}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 88 }(27,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{88}(27,·)) = \sum_{r \in \Z/88\Z} \chi_{88}(27,r) e\left(\frac{1 r + 2 r^{-1}}{88}\right) = -13.8777341788+4.5091491738i$$