# Properties

 Label 539.36 Modulus $539$ Conductor $539$ Order $35$ Real no Primitive yes Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(539, base_ring=CyclotomicField(70))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([20,56]))

pari: [g,chi] = znchar(Mod(36,539))

## Basic properties

 Modulus: $$539$$ Conductor: $$539$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$35$$ 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 539.v

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: $\Q(\zeta_{35})$ Fixed field: 35.35.73261800077965937220382205398471606200231960977600836588587975360331959691780081.1

## Values on generators

$$(199,442)$$ → $$(e\left(\frac{2}{7}\right),e\left(\frac{4}{5}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$12$$ $$13$$ $$1$$ $$1$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{16}{35}\right)$$ $$e\left(\frac{17}{35}\right)$$ $$e\left(\frac{32}{35}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{13}{35}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{8}{35}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 539 }(36,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{539}(36,\cdot)) = \sum_{r\in \Z/539\Z} \chi_{539}(36,r) e\left(\frac{2r}{539}\right) = -9.3578491906+-21.2468976212i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 539 }(36,·),\chi_{ 539 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{539}(36,\cdot),\chi_{539}(1,\cdot)) = \sum_{r\in \Z/539\Z} \chi_{539}(36,r) \chi_{539}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 539 }(36,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{539}(36,·)) = \sum_{r \in \Z/539\Z} \chi_{539}(36,r) e\left(\frac{1 r + 2 r^{-1}}{539}\right) = 30.7072360463+26.8281042019i$$