# Properties

 Label 19.18 Modulus $19$ Conductor $19$ Order $2$ Real yes Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(19, base_ring=CyclotomicField(2))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1]))

pari: [g,chi] = znchar(Mod(18,19))

## Kronecker symbol representation

sage: kronecker_character(-19)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{-19}{\bullet}\right)$$

## Basic properties

 Modulus: $$19$$ Conductor: $$19$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 19.b

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

$$2$$ → $$-1$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{-19})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 19 }(18,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{19}(18,\cdot)) = \sum_{r\in \Z/19\Z} \chi_{19}(18,r) e\left(\frac{2r}{19}\right) = -4.3588989435i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 19 }(18,·),\chi_{ 19 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{19}(18,\cdot),\chi_{19}(1,\cdot)) = \sum_{r\in \Z/19\Z} \chi_{19}(18,r) \chi_{19}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 19 }(18,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{19}(18,·)) = \sum_{r \in \Z/19\Z} \chi_{19}(18,r) e\left(\frac{1 r + 2 r^{-1}}{19}\right) = 0.0$$