Properties

 Modulus 95 Conductor 95 Order 12 Real no Primitive yes Minimal yes Parity even Orbit label 95.l

Related objects

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

sage: H = DirichletGroup(95)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3,10]))

pari: [g,chi] = znchar(Mod(12,95))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 95 Conductor = 95 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 12 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 = even Orbit label = 95.l Orbit index = 12

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

$$(77,21)$$ → $$(i,e\left(\frac{5}{6}\right))$$

Values

 -1 1 2 3 4 6 7 8 9 11 12 13 $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{12})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 95 }(12,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{95}(12,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(12,r) e\left(\frac{2r}{95}\right) = 8.7522420267+4.2893192358i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 95 }(12,·),\chi_{ 95 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{95}(12,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(12,r) \chi_{95}(1,1-r) = 1$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 95 }(12,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{95}(12,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(12,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = -4.867909666+-1.3043524638i$$