# Properties

 Modulus 65 Conductor 65 Order 4 Real no Primitive yes Minimal yes Parity even Orbit label 65.k

# Related objects

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(65)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3,1]))

pari: [g,chi] = znchar(Mod(8,65))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 65 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 Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 65.k Orbit index = 11

## Galois orbit

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(27,41)$$ → $$(-i,i)$$

## Values

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

## Related number fields

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

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 65 }(8,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{65}(8,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(8,r) e\left(\frac{2r}{65}\right) = 6.0440986817+5.3356228433i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 65 }(8,·),\chi_{ 65 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{65}(8,\cdot),\chi_{65}(1,\cdot)) = \sum_{r\in \Z/65\Z} \chi_{65}(8,r) \chi_{65}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 65 }(8,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{65}(8,·)) = \sum_{r \in \Z/65\Z} \chi_{65}(8,r) e\left(\frac{1 r + 2 r^{-1}}{65}\right) = 2.2444794081$$