Properties

 Conductor 35 Order 6 Real no Primitive no Minimal yes Parity even Orbit label 105.q

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(105)

sage: chi = H[4]

pari: [g,chi] = znchar(Mod(4,105))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 35 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 = even Orbit label = 105.q Orbit index = 17

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

$$(71,22,31)$$ → $$(1,-1,e\left(\frac{2}{3}\right))$$

Values

 -1 1 2 4 8 11 13 16 17 19 22 23 $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$
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)

$$\tau_{ a }( \chi_{ 105 }(4,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{105}(4,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(4,r) e\left(\frac{2r}{105}\right) = -0.3746818155+-5.90420304i$$

Jacobi sum

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

$$J(\chi_{ 105 }(4,·),\chi_{ 105 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{105}(4,\cdot),\chi_{105}(1,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(4,r) \chi_{105}(1,1-r) = 1$$

Kloosterman sum

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

$$K(a,b,\chi_{ 105 }(4,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{105}(4,·)) = \sum_{r \in \Z/105\Z} \chi_{105}(4,r) e\left(\frac{1 r + 2 r^{-1}}{105}\right) = -0.0$$