# Properties

 Modulus 283 Conductor 283 Order 94 Real no Primitive yes Minimal yes Parity odd Orbit label 283.f

# Learn more about

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

sage: H = DirichletGroup(283)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3]))

pari: [g,chi] = znchar(Mod(156,283))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 283 Conductor = 283 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 94 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 283.f Orbit index = 6

## Galois orbit

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

$$3$$ → $$e\left(\frac{3}{94}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{7}{94}\right)$$ $$e\left(\frac{3}{94}\right)$$ $$e\left(\frac{7}{47}\right)$$ $$e\left(\frac{41}{94}\right)$$ $$e\left(\frac{5}{47}\right)$$ $$e\left(\frac{41}{47}\right)$$ $$e\left(\frac{21}{94}\right)$$ $$e\left(\frac{3}{47}\right)$$ $$e\left(\frac{24}{47}\right)$$ $$e\left(\frac{40}{47}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{47})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 283 }(156,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{283}(156,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(156,r) e\left(\frac{2r}{283}\right) = -12.7237839631+11.0047863069i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 283 }(156,·),\chi_{ 283 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{283}(156,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(156,r) \chi_{283}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 283 }(156,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{283}(156,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(156,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = -5.8965349076+24.7429113085i$$