Properties

 Label 8470.5939 Modulus $8470$ Conductor $4235$ Order $66$ Real no Primitive no Minimal yes Parity even

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

sage: H = DirichletGroup(8470, base_ring=CyclotomicField(66))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([33,11,45]))

pari: [g,chi] = znchar(Mod(5939,8470))

Basic properties

 Modulus: $$8470$$ Conductor: $$4235$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$66$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{4235}(1704,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 8470.cm

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: $$\Q(\zeta_{33})$$ Fixed field: Number field defined by a degree 66 polynomial

Values on generators

$$(6777,6051,7141)$$ → $$(-1,e\left(\frac{1}{6}\right),e\left(\frac{15}{22}\right))$$

Values

 $$-1$$ $$1$$ $$3$$ $$9$$ $$13$$ $$17$$ $$19$$ $$23$$ $$27$$ $$29$$ $$31$$ $$37$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{31}{66}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 8470 }(5939,a) \;$$ at $$\;a =$$ e.g. 2