Properties

 Modulus 68 Conductor 68 Order 4 Real no Primitive yes Minimal yes Parity odd Orbit label 68.f

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

sage: H = DirichletGroup(68)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([2,1]))

pari: [g,chi] = znchar(Mod(47,68))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 68 Conductor = 68 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 Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 68.f Orbit index = 6

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

$$(35,37)$$ → $$(-1,i)$$

Values

 -1 1 3 5 7 9 11 13 15 19 21 23 $$-1$$ $$1$$ $$-i$$ $$i$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$1$$ $$1$$ $$i$$
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 68 }(47,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{68}(47,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(47,r) e\left(\frac{r}{34}\right) = 0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 68 }(47,·),\chi_{ 68 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{68}(47,\cdot),\chi_{68}(1,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(47,r) \chi_{68}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 68 }(47,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{68}(47,·)) = \sum_{r \in \Z/68\Z} \chi_{68}(47,r) e\left(\frac{1 r + 2 r^{-1}}{68}\right) = -13.6467382206$$