# Properties

 Label 3549.731 Modulus $3549$ Conductor $3549$ Order $78$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(3549, base_ring=CyclotomicField(78))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([39,13,56]))

pari: [g,chi] = znchar(Mod(731,3549))

## Basic properties

 Modulus: $$3549$$ Conductor: $$3549$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$78$$ 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 3549.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: $\Q(\zeta_{39})$ Fixed field: Number field defined by a degree 78 polynomial

## Values on generators

$$(1184,1522,3382)$$ → $$(-1,e\left(\frac{1}{6}\right),e\left(\frac{28}{39}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$16$$ $$17$$ $$19$$ $$20$$ $$1$$ $$1$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$-1$$ $$e\left(\frac{35}{39}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 3549 }(731,a) \;$$ at $$\;a =$$ e.g. 2