# Properties

 Label 6243.6242 Modulus $6243$ Conductor $6243$ Order $2$ Real yes Primitive yes Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(6243, base_ring=CyclotomicField(2))

M = H._module

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

pari: [g,chi] = znchar(Mod(6242,6243))

## Kronecker symbol representation

sage: kronecker_character(-6243)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{-6243}{\bullet}\right)$$

## Basic properties

 Modulus: $$6243$$ Conductor: $$6243$$ 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 6243.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: $$\Q(\sqrt{-6243})$$

## Values on generators

$$(4163,4165)$$ → $$(-1,-1)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$\chi_{ 6243 }(6242, a)$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 6243 }(6242,a) \;$$ at $$\;a =$$ e.g. 2