# Properties

 Label 5.2 Modulus $5$ Conductor $5$ Order $4$ Real no Primitive yes Minimal yes Parity odd

# Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5, base_ring=CyclotomicField(4))

M = H._module

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

pari: [g,chi] = znchar(Mod(2,5))

## 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()

order = charorder(g,chi)

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

## Related number fields

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

## Values on generators

$$2$$ → $$i$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$\chi_{ 5 }(2, a)$$ $$-1$$ $$1$$ $$i$$ $$-i$$
sage: chi.jacobi_sum(n)

$$\chi_{ 5 }(2,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 5 }(2,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 5 }(2,·),\chi_{ 5 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 5 }(2,·)) \;$$ at $$\; a,b =$$ e.g. 1,2

# Additional information

This is the first example of a Dirichlet character whose values do not all lie in the field of rational numbers.

This also makes it the first example of a Dirichlet character whose Galois orbit is nontrivial.