# Properties

 Label 845.699 Modulus $845$ Conductor $65$ Order $6$ Real no Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(845, base_ring=CyclotomicField(6))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3,5]))

pari: [g,chi] = znchar(Mod(699,845))

## Basic properties

 Modulus: $$845$$ Conductor: $$65$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$6$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{65}(49,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 845.l

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: $$\Q(\sqrt{-3})$$ Fixed field: 6.6.46411625.1

## Values on generators

$$(677,171)$$ → $$(-1,e\left(\frac{5}{6}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$14$$ $$\chi_{ 845 }(699, a)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 845 }(699,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 845 }(699,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 845 }(699,·),\chi_{ 845 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 845 }(699,·)) \;$$ at $$\; a,b =$$ e.g. 1,2