# Properties

 Conductor 193 Order 64 Real no Primitive yes Minimal yes Parity odd Orbit label 193.l

# 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(193)

sage: chi = H[164]

pari: [g,chi] = znchar(Mod(164,193))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 193 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 64 Real = no 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 = 193.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

$$5$$ → $$e\left(\frac{9}{64}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{59}{64}\right)$$ $$e\left(\frac{47}{64}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{64})$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 193 }(164,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{193}(164,\cdot)) = \sum_{r\in \Z/193\Z} \chi_{193}(164,r) e\left(\frac{2r}{193}\right) = 2.0398558317+-13.7418698941i$$

## Jacobi sum

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

$$J(\chi_{ 193 }(164,·),\chi_{ 193 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{193}(164,\cdot),\chi_{193}(1,\cdot)) = \sum_{r\in \Z/193\Z} \chi_{193}(164,r) \chi_{193}(1,1-r) = -1$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 193 }(164,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{193}(164,·)) = \sum_{r \in \Z/193\Z} \chi_{193}(164,r) e\left(\frac{1 r + 2 r^{-1}}{193}\right) = 14.7304706974+17.9491304784i$$