# Properties

 Conductor 65 Order 4 Real No Primitive Yes Parity Odd Orbit Label 65.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(65)
sage: chi = H[44]
pari: [g,chi] = znchar(Mod(44,65))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 65 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 4 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 = 65.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 ]

## Values on generators

$$(27,41)$$ → $$(-1,-i)$$

## Values

 -1 1 2 3 4 6 7 8 9 11 12 14 $$-1$$ $$1$$ $$i$$ $$-1$$ $$-1$$ $$-i$$ $$-i$$ $$-i$$ $$1$$ $$i$$ $$1$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(i)$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 65 }(44,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{65}(44,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(44,r) e\left(\frac{2r}{65}\right) = 7.7163226064+2.3363144981i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 65 }(44,·),\chi_{ 65 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{65}(44,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(44,r) \chi_{65}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 65 }(44,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{65}(44,·)) = \sum_{r \in \Z/65\Z} \chi_{65}(44,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right) = 0.0$$