# Properties

 Label 4235.899 Modulus $4235$ Conductor $4235$ Order $330$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(4235, base_ring=CyclotomicField(330))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([165,55,309]))

pari: [g,chi] = znchar(Mod(899,4235))

## Basic properties

 Modulus: $$4235$$ Conductor: $$4235$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$330$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no 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 4235.dj

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $\Q(\zeta_{165})$ Fixed field: Number field defined by a degree 330 polynomial (not computed)

## Values on generators

$$(2542,1816,2906)$$ → $$(-1,e\left(\frac{1}{6}\right),e\left(\frac{103}{110}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$8$$ $$9$$ $$12$$ $$13$$ $$16$$ $$17$$ $$1$$ $$1$$ $$e\left(\frac{127}{165}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{89}{165}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{13}{165}\right)$$ $$e\left(\frac{181}{330}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 4235 }(899,a) \;$$ at $$\;a =$$ e.g. 2