Properties

 Conductor 32 Order 8 Real no Primitive no Minimal yes Parity odd Orbit label 160.w

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(160)

sage: chi = H[11]

pari: [g,chi] = znchar(Mod(11,160))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 32 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 8 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 = 160.w Orbit index = 23

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

$$(31,101,97)$$ → $$(-1,e\left(\frac{5}{8}\right),1)$$

Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{8})$$

Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 160 }(11,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{160}(11,\cdot)) = \sum_{r\in \Z/160\Z} \chi_{160}(11,r) e\left(\frac{r}{80}\right) = -0.0$$

Jacobi sum

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

$$J(\chi_{ 160 }(11,·),\chi_{ 160 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{160}(11,\cdot),\chi_{160}(1,\cdot)) = \sum_{r\in \Z/160\Z} \chi_{160}(11,r) \chi_{160}(1,1-r) = 0$$

Kloosterman sum

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

$$K(a,b,\chi_{ 160 }(11,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{160}(11,·)) = \sum_{r \in \Z/160\Z} \chi_{160}(11,r) e\left(\frac{1 r + 2 r^{-1}}{160}\right) = -5.8138487107+3.8846895126i$$