# Properties

 Label 4008.1403 Modulus $4008$ Conductor $4008$ Order $166$ 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(4008)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([83,83,83,165]))

pari: [g,chi] = znchar(Mod(1403,4008))

## Basic properties

 Modulus: $$4008$$ Conductor: $$4008$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$166$$ 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 4008.be

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

$$(3007,2005,1337,673)$$ → $$(-1,-1,-1,e\left(\frac{165}{166}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$-1$$ $$1$$ $$e\left(\frac{165}{166}\right)$$ $$e\left(\frac{131}{166}\right)$$ $$e\left(\frac{55}{166}\right)$$ $$e\left(\frac{73}{83}\right)$$ $$e\left(\frac{15}{83}\right)$$ $$e\left(\frac{54}{83}\right)$$ $$e\left(\frac{67}{166}\right)$$ $$e\left(\frac{82}{83}\right)$$ $$e\left(\frac{8}{83}\right)$$ $$e\left(\frac{159}{166}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{83})$ Fixed field: Number field defined by a degree 166 polynomial