# Properties

 Label 21.20 Modulus $21$ Conductor $21$ Order $2$ Real yes Primitive yes Minimal yes Parity even

# Learn more

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

H = DirichletGroup(21, base_ring=CyclotomicField(2))

M = H._module

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

pari: [g,chi] = znchar(Mod(20,21))

## Kronecker symbol representation

sage: kronecker_character(21)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{21}{\bullet}\right)$$

## Basic properties

 Modulus: $$21$$ Conductor: $$21$$ 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: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 21.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: $$\Q$$ Fixed field: $$\Q(\sqrt{21})$$

## Values on generators

$$(8,10)$$ → $$(-1,-1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 21 }(20, a)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 21 }(20,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 21 }(20,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 21 }(20,·),\chi_{ 21 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 21 }(20,·)) \;$$ at $$\; a,b =$$ e.g. 1,2