# Properties

 Modulus 210 Conductor 7 Order 6 Real no Primitive no Minimal yes Parity odd Orbit label 210.o

# 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([0,0,5]))

pari: [g,chi] = znchar(Mod(61,210))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 210 Conductor = 7 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 6 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 = odd Orbit label = 210.o Orbit index = 15

## 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

$$(71,127,31)$$ → $$(1,1,e\left(\frac{5}{6}\right))$$

## Values

 -1 1 11 13 17 19 23 29 31 37 41 43 $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{3})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 210 }(61,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{210}(61,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(61,r) e\left(\frac{r}{105}\right) = 2.4401333583+1.0226187919i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 210 }(61,·),\chi_{ 210 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{210}(61,\cdot),\chi_{210}(1,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(61,r) \chi_{210}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 210 }(61,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{210}(61,·)) = \sum_{r \in \Z/210\Z} \chi_{210}(61,r) e\left(\frac{1 r + 2 r^{-1}}{210}\right) = 2.3166781499+1.3375347535i$$