# Properties

 Label 712.131 Modulus $712$ Conductor $712$ Order $44$ 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(712, base_ring=CyclotomicField(44))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([22,22,5]))

pari: [g,chi] = znchar(Mod(131,712))

## Basic properties

 Modulus: $$712$$ Conductor: $$712$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ 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 712.y

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(535,357,537)$$ → $$(-1,-1,e\left(\frac{5}{44}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$-1$$ $$1$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{9}{11}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 712 }(131,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 712 }(131,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 712 }(131,·),\chi_{ 712 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 712 }(131,·)) \;$$ at $$\; a,b =$$ e.g. 1,2