Properties

 Modulus 78 Conductor 13 Order 4 Real no Primitive no Minimal yes Parity odd Orbit label 78.f

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

sage: H = DirichletGroup(78)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,1]))

pari: [g,chi] = znchar(Mod(73,78))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 78 Conductor = 13 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 = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 78.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

$$(53,67)$$ → $$(1,i)$$

Values

 -1 1 5 7 11 17 19 23 25 29 31 35 $$-1$$ $$1$$ $$i$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$-1$$ $$-1$$ $$1$$ $$i$$ $$1$$
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)

$$\tau_{ a }( \chi_{ 78 }(73,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{78}(73,\cdot)) = \sum_{r\in \Z/78\Z} \chi_{78}(73,r) e\left(\frac{r}{39}\right) = 3.4508443768+-1.0448316069i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 78 }(73,·),\chi_{ 78 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{78}(73,\cdot),\chi_{78}(1,\cdot)) = \sum_{r\in \Z/78\Z} \chi_{78}(73,r) \chi_{78}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 78 }(73,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{78}(73,·)) = \sum_{r \in \Z/78\Z} \chi_{78}(73,r) e\left(\frac{1 r + 2 r^{-1}}{78}\right) = -8.6681631243+8.6681631243i$$