# Properties

 Label 3.2 Modulus $3$ Conductor $3$ 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(3)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(2,3))

## Kronecker symbol representation

sage: kronecker_character(-3)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{-3}{\bullet}\right)$$

## Basic properties

 Modulus: $$3$$ Conductor: $$3$$ 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 3.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

 $$-1$$ $$1$$ $$-1$$ $$1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{-3})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 3 }(2,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{3}(2,\cdot)) = \sum_{r\in \Z/3\Z} \chi_{3}(2,r) e\left(\frac{2r}{3}\right) = -1.7320508076i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 3 }(2,·),\chi_{ 3 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{3}(2,\cdot),\chi_{3}(1,\cdot)) = \sum_{r\in \Z/3\Z} \chi_{3}(2,r) \chi_{3}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 3 }(2,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{3}(2,·)) = \sum_{r \in \Z/3\Z} \chi_{3}(2,r) e\left(\frac{1 r + 2 r^{-1}}{3}\right) = 0.0$$

This is the first Dirichlet character taking a value other than $1$. It is also the first Dirichlet character with odd parity.