# Properties

 Label 693.520 Modulus $693$ Conductor $693$ Order $15$ 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(693, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([20,10,24]))

pari: [g,chi] = znchar(Mod(520,693))

## Basic properties

 Modulus: $$693$$ Conductor: $$693$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$15$$ 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 693.bx

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: $$\Q(\zeta_{15})$$ Fixed field: 15.15.3091133177133909578645502426129.1

## Values on generators

$$(155,199,442)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right),e\left(\frac{4}{5}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$13$$ $$16$$ $$17$$ $$19$$ $$20$$ $$1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 693 }(520,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{693}(520,\cdot)) = \sum_{r\in \Z/693\Z} \chi_{693}(520,r) e\left(\frac{2r}{693}\right) = -9.9759576377+24.3614504743i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 693 }(520,·),\chi_{ 693 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{693}(520,\cdot),\chi_{693}(1,\cdot)) = \sum_{r\in \Z/693\Z} \chi_{693}(520,r) \chi_{693}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 693 }(520,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{693}(520,·)) = \sum_{r \in \Z/693\Z} \chi_{693}(520,r) e\left(\frac{1 r + 2 r^{-1}}{693}\right) = -12.7997307521+-5.698807295i$$