# Properties

 Conductor 7 Order 6 Real No Primitive No Parity Odd Orbit Label 56.o

# 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(56)
sage: chi = H[33]
pari: [g,chi] = znchar(Mod(33,56))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 7 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 6 Real = No sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = No sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 56.o Orbit index = 15

## 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

$$(15,29,17)$$ → $$(1,1,e\left(\frac{5}{6}\right))$$

## Values

 -1 1 3 5 9 11 13 15 17 19 23 25 $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{3})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 56 }(33,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{56}(33,\cdot)) = \sum_{r\in \Z/56\Z} \chi_{56}(33,r) e\left(\frac{r}{28}\right) = 0.0$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 56 }(33,·),\chi_{ 56 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{56}(33,\cdot),\chi_{56}(1,\cdot)) = \sum_{r\in \Z/56\Z} \chi_{56}(33,r) \chi_{56}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 56 }(33,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{56}(33,·)) = \sum_{r \in \Z/56\Z} \chi_{56}(33,r) e\left(\frac{1 r + 2 r^{-1}}{56}\right) = -0.0$$