# Properties

 Conductor 203 Order 84 Real No Primitive Yes Parity Odd Orbit Label 203.w

# 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(203)
sage: chi = H[18]
pari: [g,chi] = znchar(Mod(18,203))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 203 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 84 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 = 203.w Orbit index = 23

## 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

$$(59,176)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{11}{28}\right))$$

## Values

 -1 1 2 3 4 5 6 8 9 10 11 12 $$-1$$ $$1$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{1}{12}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{84})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 203 }(18,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{203}(18,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(18,r) e\left(\frac{2r}{203}\right) = 4.7476612328+13.4335294252i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 203 }(18,·),\chi_{ 203 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{203}(18,\cdot),\chi_{203}(1,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(18,r) \chi_{203}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 203 }(18,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{203}(18,·)) = \sum_{r \in \Z/203\Z} \chi_{203}(18,r) e\left(\frac{1 r + 2 r^{-1}}{203}\right) = -18.8396735918+-16.2128500856i$$