Properties

 Modulus 96 Conductor 96 Order 8 Real no Primitive yes Minimal yes Parity odd Orbit label 96.p

Related objects

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

sage: H = DirichletGroup(96)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,7,4]))

pari: [g,chi] = znchar(Mod(77,96))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 96 Conductor = 96 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 = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 96.p Orbit index = 16

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

$$(31,37,65)$$ → $$(1,e\left(\frac{7}{8}\right),-1)$$

Values

 -1 1 5 7 11 13 17 19 23 25 29 31 $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 96 }(77,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{96}(77,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(77,r) e\left(\frac{r}{48}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 96 }(77,·),\chi_{ 96 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{96}(77,\cdot),\chi_{96}(1,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(77,r) \chi_{96}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 96 }(77,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{96}(77,·)) = \sum_{r \in \Z/96\Z} \chi_{96}(77,r) e\left(\frac{1 r + 2 r^{-1}}{96}\right) = 0.0$$