# Properties

 Label 1728.571 Modulus $1728$ Conductor $1728$ Order $144$ 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(1728, base_ring=CyclotomicField(144))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([72,9,16]))

pari: [g,chi] = znchar(Mod(571,1728))

## Basic properties

 Modulus: $$1728$$ Conductor: $$1728$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$144$$ 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 1728.cg

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: $\Q(\zeta_{144})$ Fixed field: Number field defined by a degree 144 polynomial (not computed)

## Values on generators

$$(703,325,1217)$$ → $$(-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{9}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$\chi_{ 1728 }(571, a)$$ $$-1$$ $$1$$ $$e\left(\frac{89}{144}\right)$$ $$e\left(\frac{65}{72}\right)$$ $$e\left(\frac{37}{144}\right)$$ $$e\left(\frac{119}{144}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{115}{144}\right)$$ $$e\left(\frac{2}{9}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1728 }(571,a) \;$$ at $$\;a =$$ e.g. 2