# Properties

 Label 571.394 Modulus $571$ Conductor $571$ Order $285$ 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(571, base_ring=CyclotomicField(570))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([346]))

pari: [g,chi] = znchar(Mod(394,571))

## Basic properties

 Modulus: $$571$$ Conductor: $$571$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$285$$ 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 571.o

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: $\Q(\zeta_{285})$ Fixed field: Number field defined by a degree 285 polynomial (not computed)

## Values on generators

$$3$$ → $$e\left(\frac{173}{285}\right)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$\chi_{ 571 }(394, a)$$ $$1$$ $$1$$ $$e\left(\frac{47}{57}\right)$$ $$e\left(\frac{173}{285}\right)$$ $$e\left(\frac{37}{57}\right)$$ $$e\left(\frac{46}{285}\right)$$ $$e\left(\frac{41}{95}\right)$$ $$e\left(\frac{21}{95}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{61}{285}\right)$$ $$e\left(\frac{281}{285}\right)$$ $$e\left(\frac{194}{285}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 571 }(394,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 571 }(394,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 571 }(394,·),\chi_{ 571 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 571 }(394,·)) \;$$ at $$\; a,b =$$ e.g. 1,2