# Properties

 Label 5.3 Modulus $5$ Conductor $5$ Order $4$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(5)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3]))

pari: [g,chi] = znchar(Mod(3,5))

## Basic properties

 Modulus: $$5$$ Conductor: $$5$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no 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 5.c

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

$$2$$ → $$-i$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$-1$$ $$1$$ $$-i$$ $$i$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\sqrt{-1})$$ Fixed field: $$\Q(\zeta_{5})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 5 }(3,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{5}(3,\cdot)) = \sum_{r\in \Z/5\Z} \chi_{5}(3,r) e\left(\frac{2r}{5}\right) = -1.9021130326+1.1755705046i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 5 }(3,·),\chi_{ 5 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{5}(3,\cdot),\chi_{5}(1,\cdot)) = \sum_{r\in \Z/5\Z} \chi_{5}(3,r) \chi_{5}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 5 }(3,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{5}(3,·)) = \sum_{r \in \Z/5\Z} \chi_{5}(3,r) e\left(\frac{1 r + 2 r^{-1}}{5}\right) = -1.1755705046+-1.1755705046i$$