Properties

 Label 191.14 Modulus $191$ Conductor $191$ Order $38$ Real no Primitive yes Minimal yes Parity odd

Related objects

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

sage: H = DirichletGroup(191)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([5]))

pari: [g,chi] = znchar(Mod(14,191))

Basic properties

 Modulus: $$191$$ Conductor: $$191$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$38$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 191.f

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

$$19$$ → $$e\left(\frac{5}{38}\right)$$

Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$-1$$ $$1$$ $$e\left(\frac{15}{19}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{1}{19}\right)$$ $$-1$$ $$e\left(\frac{7}{19}\right)$$ $$e\left(\frac{10}{19}\right)$$ $$e\left(\frac{7}{19}\right)$$ $$e\left(\frac{7}{38}\right)$$
 value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 191 }(14,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{191}(14,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(14,r) e\left(\frac{2r}{191}\right) = -10.4990524173+8.9872074828i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 191 }(14,·),\chi_{ 191 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{191}(14,\cdot),\chi_{191}(1,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(14,r) \chi_{191}(1,1-r) = -1$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 191 }(14,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{191}(14,·)) = \sum_{r \in \Z/191\Z} \chi_{191}(14,r) e\left(\frac{1 r + 2 r^{-1}}{191}\right) = -3.622320982+-4.6539580928i$$