Properties

 Conductor 21 Order 6 Real No Primitive No Parity Even Orbit Label 336.bc

Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(336)

sage: chi = H[17]

pari: [g,chi] = znchar(Mod(17,336))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 21 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 sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = Even Orbit label = 336.bc Orbit index = 29

Galois orbit

sage: chi.sage_character().galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Values on generators

$$(127,85,113,241)$$ → $$(1,1,-1,e\left(\frac{1}{6}\right))$$

Values

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

Related number fields

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

Gauss sum

sage: chi.sage_character().gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 336 }(17,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{336}(17,\cdot)) = \sum_{r\in \Z/336\Z} \chi_{336}(17,r) e\left(\frac{r}{168}\right) = 0.0$$

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)

$$J(\chi_{ 336 }(17,·),\chi_{ 336 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{336}(17,\cdot),\chi_{336}(1,\cdot)) = \sum_{r\in \Z/336\Z} \chi_{336}(17,r) \chi_{336}(1,1-r) = 0$$

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)

$$K(a,b,\chi_{ 336 }(17,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{336}(17,·)) = \sum_{r \in \Z/336\Z} \chi_{336}(17,r) e\left(\frac{1 r + 2 r^{-1}}{336}\right) = 0.0$$