# Properties

 Label 1089.923 Modulus $1089$ Conductor $1089$ Order $66$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(1089, base_ring=CyclotomicField(66))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([55,51]))

pari: [g,chi] = znchar(Mod(923,1089))

## Basic properties

 Modulus: $$1089$$ Conductor: $$1089$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$66$$ 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 1089.y

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

$$(848,244)$$ → $$(e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$ $$1$$ $$1$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{4}{11}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1089 }(923,a) \;$$ at $$\;a =$$ e.g. 2