Properties

 Conductor 80 Order 4 Real No Primitive Yes Parity Odd Orbit Label 80.i

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(80)

sage: chi = H[37]

pari: [g,chi] = znchar(Mod(37,80))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 80 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 4 Real = No sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = Odd Orbit label = 80.i Orbit index = 9

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

$$(31,21,17)$$ → $$(1,i,i)$$

Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$-1$$ $$1$$ $$-1$$ $$-i$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$i$$ $$i$$ $$-1$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(i)$$

Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 80 }(37,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{80}(37,\cdot)) = \sum_{r\in \Z/80\Z} \chi_{80}(37,r) e\left(\frac{r}{40}\right) = 0.0$$

Jacobi sum

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

$$J(\chi_{ 80 }(37,·),\chi_{ 80 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{80}(37,\cdot),\chi_{80}(1,\cdot)) = \sum_{r\in \Z/80\Z} \chi_{80}(37,r) \chi_{80}(1,1-r) = 0$$

Kloosterman sum

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

$$K(a,b,\chi_{ 80 }(37,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{80}(37,·)) = \sum_{r \in \Z/80\Z} \chi_{80}(37,r) e\left(\frac{1 r + 2 r^{-1}}{80}\right) = -2.5448566901+6.1438275356i$$