# Properties

 Label 161.25 Modulus $161$ Conductor $161$ Order $33$ 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(161, base_ring=CyclotomicField(66))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([44,6]))

pari: [g,chi] = znchar(Mod(25,161))

## Basic properties

 Modulus: $$161$$ Conductor: $$161$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$33$$ 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 161.m

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: $$\Q(\zeta_{33})$$ Fixed field: 33.33.277966181338944111003326058293667039541136678070715028736001.1

## Values on generators

$$(24,120)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$1$$ $$1$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 161 }(25,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 161 }(25,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 161 }(25,·),\chi_{ 161 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 161 }(25,·)) \;$$ at $$\; a,b =$$ e.g. 1,2