# Properties

 Modulus 168 Conductor 168 Order 6 Real no Primitive yes Minimal yes Parity even Orbit label 168.ba

# Related objects

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

sage: H = DirichletGroup(168)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3,3,5]))

pari: [g,chi] = znchar(Mod(5,168))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 168 Conductor = 168 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 = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 168.ba Orbit index = 27

## 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

$$(127,85,113,73)$$ → $$(1,-1,-1,e\left(\frac{5}{6}\right))$$

## Values

 -1 1 5 11 13 17 19 23 25 29 31 37 $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
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)

$$\tau_{ a }( \chi_{ 168 }(5,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{168}(5,\cdot)) = \sum_{r\in \Z/168\Z} \chi_{168}(5,r) e\left(\frac{r}{84}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 168 }(5,·),\chi_{ 168 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{168}(5,\cdot),\chi_{168}(1,\cdot)) = \sum_{r\in \Z/168\Z} \chi_{168}(5,r) \chi_{168}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 168 }(5,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{168}(5,·)) = \sum_{r \in \Z/168\Z} \chi_{168}(5,r) e\left(\frac{1 r + 2 r^{-1}}{168}\right) = -0.0$$