# Properties

 Conductor 4 Order 2 Real Yes Primitive No Parity Odd Orbit Label 72.g

# 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(72)
sage: chi = H[55]
pari: [g,chi] = znchar(Mod(55,72))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 4 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 2 Real = Yes 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 = 72.g Orbit index = 7

## 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 ]

## Inducingprimitive character

$$\chi_{4}(3,\cdot)$$ = $$\displaystyle\left(\frac{-4}{\bullet}\right)$$

## Values on generators

$$(55,37,65)$$ → $$(-1,1,1)$$

## Values

 -1 1 5 7 11 13 17 19 23 25 29 31 $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 72 }(55,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{72}(55,\cdot)) = \sum_{r\in \Z/72\Z} \chi_{72}(55,r) e\left(\frac{r}{36}\right) = -0.0$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 72 }(55,·),\chi_{ 72 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{72}(55,\cdot),\chi_{72}(1,\cdot)) = \sum_{r\in \Z/72\Z} \chi_{72}(55,r) \chi_{72}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 72 }(55,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{72}(55,·)) = \sum_{r \in \Z/72\Z} \chi_{72}(55,r) e\left(\frac{1 r + 2 r^{-1}}{72}\right) = -0.0$$