# Properties

 Label 1078.579 Modulus $1078$ Conductor $539$ Order $210$ Real no Primitive no Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(1078, base_ring=CyclotomicField(210))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([115,147]))

pari: [g,chi] = znchar(Mod(579,1078))

## Basic properties

 Modulus: $$1078$$ Conductor: $$539$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$210$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{539}(40,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1078.bf

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: $\Q(\zeta_{105})$ Fixed field: Number field defined by a degree 210 polynomial (not computed)

## Values on generators

$$(199,981)$$ → $$(e\left(\frac{23}{42}\right),e\left(\frac{7}{10}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$27$$ $$\chi_{ 1078 }(579, a)$$ $$1$$ $$1$$ $$e\left(\frac{31}{210}\right)$$ $$e\left(\frac{143}{210}\right)$$ $$e\left(\frac{31}{105}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{29}{35}\right)$$ $$e\left(\frac{104}{105}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{38}{105}\right)$$ $$e\left(\frac{31}{70}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1078 }(579,a) \;$$ at $$\;a =$$ e.g. 2