# Properties

 Label 8470.1789 Modulus $8470$ Conductor $4235$ Order $330$ Real no Primitive no Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(8470, base_ring=CyclotomicField(330))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([165,220,141]))

pari: [g,chi] = znchar(Mod(1789,8470))

## Basic properties

 Modulus: $$8470$$ Conductor: $$4235$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$330$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{4235}(1789,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 8470.dk

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

$$(6777,6051,7141)$$ → $$(-1,e\left(\frac{2}{3}\right),e\left(\frac{47}{110}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$9$$ $$13$$ $$17$$ $$19$$ $$23$$ $$27$$ $$29$$ $$31$$ $$37$$ $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{17}{165}\right)$$ $$e\left(\frac{263}{330}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{68}{165}\right)$$ $$e\left(\frac{257}{330}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 8470 }(1789,a) \;$$ at $$\;a =$$ e.g. 2