# Properties

 Label 160.29 Modulus $160$ Conductor $160$ Order $8$ 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(160, base_ring=CyclotomicField(8))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3,4]))

pari: [g,chi] = znchar(Mod(29,160))

## Basic properties

 Modulus: $$160$$ Conductor: $$160$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ 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 160.z

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: $$\Q(\zeta_{8})$$ Fixed field: 8.8.1342177280000.1

## Values on generators

$$(31,101,97)$$ → $$(1,e\left(\frac{3}{8}\right),-1)$$

## Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 160 }(29,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 160 }(29,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 160 }(29,·),\chi_{ 160 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 160 }(29,·)) \;$$ at $$\; a,b =$$ e.g. 1,2