# Properties

 Label 65.57 Modulus $65$ Conductor $65$ Order $4$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(65, base_ring=CyclotomicField(4))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(57,65))

## Basic properties

 Modulus: $$65$$ Conductor: $$65$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 65.k

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\sqrt{-1})$$ Fixed field: 4.4.274625.1

## Values on generators

$$(27,41)$$ → $$(i,-i)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$14$$ $$\chi_{ 65 }(57, a)$$ $$1$$ $$1$$ $$1$$ $$-i$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 65 }(57,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 65 }(57,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 65 }(57,·),\chi_{ 65 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 65 }(57,·)) \;$$ at $$\; a,b =$$ e.g. 1,2