# Properties

 Modulus 210 Conductor 3 Order 2 Real yes Primitive no Minimal yes Parity odd Orbit label 210.e

# Related objects

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

sage: H = DirichletGroup(210)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,0,0]))

pari: [g,chi] = znchar(Mod(71,210))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 210 Conductor = 3 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 2 Real = yes 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 = odd Orbit label = 210.e Orbit index = 5

## 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

$$(71,127,31)$$ → $$(-1,1,1)$$

## Values

 -1 1 11 13 17 19 23 29 31 37 41 43 $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 210 }(71,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{210}(71,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(71,r) e\left(\frac{r}{105}\right) = -1.7320508076i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 210 }(71,·),\chi_{ 210 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{210}(71,\cdot),\chi_{210}(1,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(71,r) \chi_{210}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 210 }(71,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{210}(71,·)) = \sum_{r \in \Z/210\Z} \chi_{210}(71,r) e\left(\frac{1 r + 2 r^{-1}}{210}\right) = 0.0$$