# Properties

 Label 1287.223 Modulus $1287$ Conductor $1287$ Order $60$ Real no Primitive yes Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(1287, base_ring=CyclotomicField(60))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([40,48,5]))

pari: [g,chi] = znchar(Mod(223,1287))

## Basic properties

 Modulus: $$1287$$ Conductor: $$1287$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$60$$ 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 1287.ep

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: $$\Q(\zeta_{60})$$ Fixed field: Number field defined by a degree 60 polynomial

## Values on generators

$$(1145,937,496)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right),e\left(\frac{1}{12}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$14$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 1287 }(223, a)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{49}{60}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1287 }(223,a) \;$$ at $$\;a =$$ e.g. 2