# Properties

 Label 4008.3005 Modulus $4008$ Conductor $4008$ Order $2$ Real yes Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(4008)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,1,1,1]))

pari: [g,chi] = znchar(Mod(3005,4008))

## Kronecker symbol representation

sage: kronecker_character(4008)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{4008}{\bullet}\right)$$

## Basic properties

 Modulus: $$4008$$ Conductor: $$4008$$ 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 4008.i

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

$$(3007,2005,1337,673)$$ → $$(1,-1,-1,-1)$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{1002})$$