# Properties

 Label 385.38 Modulus $385$ Conductor $385$ Order $60$ 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(385, base_ring=CyclotomicField(60))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([45,10,24]))

pari: [g,chi] = znchar(Mod(38,385))

## Basic properties

 Modulus: $$385$$ Conductor: $$385$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$60$$ 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 385.bu

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: $$\Q(\zeta_{60})$$ Fixed field: Number field defined by a degree 60 polynomial

## Values on generators

$$(232,276,211)$$ → $$(-i,e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$8$$ $$9$$ $$12$$ $$13$$ $$16$$ $$17$$ $$\chi_{ 385 }(38, a)$$ $$1$$ $$1$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{31}{60}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 385 }(38,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 385 }(38,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 385 }(38,·),\chi_{ 385 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 385 }(38,·)) \;$$ at $$\; a,b =$$ e.g. 1,2