# Properties

 Modulus 62 Conductor 31 Order 5 Real no Primitive no Minimal yes Parity even Orbit label 62.d

# Related objects

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

sage: H = DirichletGroup(62)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1]))

pari: [g,chi] = znchar(Mod(47,62))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 62 Conductor = 31 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 5 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 62.d Orbit index = 4

## 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

$$3$$ → $$e\left(\frac{1}{5}\right)$$

## Values

 -1 1 3 5 7 9 11 13 15 17 19 21 $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{5})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 62 }(47,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{62}(47,\cdot)) = \sum_{r\in \Z/62\Z} \chi_{62}(47,r) e\left(\frac{r}{31}\right) = 4.5524166695+-3.2055424607i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 62 }(47,·),\chi_{ 62 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{62}(47,\cdot),\chi_{62}(1,\cdot)) = \sum_{r\in \Z/62\Z} \chi_{62}(47,r) \chi_{62}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 62 }(47,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{62}(47,·)) = \sum_{r \in \Z/62\Z} \chi_{62}(47,r) e\left(\frac{1 r + 2 r^{-1}}{62}\right) = 4.7579495475+-3.4568526924i$$