# Properties

 Label 259.9 Modulus $259$ Conductor $259$ Order $9$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(259, base_ring=CyclotomicField(18))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([6,8]))

pari: [g,chi] = znchar(Mod(9,259))

## Basic properties

 Modulus: $$259$$ Conductor: $$259$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$9$$ 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 259.y

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(38,113)$$ → $$(e\left(\frac{1}{3}\right),e\left(\frac{4}{9}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{9})$$ Fixed field: 9.9.413239695274351729.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 259 }(9,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{259}(9,\cdot)) = \sum_{r\in \Z/259\Z} \chi_{259}(9,r) e\left(\frac{2r}{259}\right) = 13.2911888533+-9.0743759491i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 259 }(9,·),\chi_{ 259 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{259}(9,\cdot),\chi_{259}(1,\cdot)) = \sum_{r\in \Z/259\Z} \chi_{259}(9,r) \chi_{259}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 259 }(9,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{259}(9,·)) = \sum_{r \in \Z/259\Z} \chi_{259}(9,r) e\left(\frac{1 r + 2 r^{-1}}{259}\right) = -9.2853623452+-3.379595508i$$