Properties

Conductor 283
Order 47
Real No
Primitive Yes
Parity Even
Orbit Label 283.e

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(283)
sage: chi = H[134]
pari: [g,chi] = znchar(Mod(134,283))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 283
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 47
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 283.e
Orbit index = 5

Galois orbit

sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

\(\chi_{283}(4,\cdot)\) \(\chi_{283}(15,\cdot)\) \(\chi_{283}(16,\cdot)\) \(\chi_{283}(29,\cdot)\) \(\chi_{283}(38,\cdot)\) \(\chi_{283}(42,\cdot)\) \(\chi_{283}(51,\cdot)\) \(\chi_{283}(54,\cdot)\) \(\chi_{283}(60,\cdot)\) \(\chi_{283}(61,\cdot)\) \(\chi_{283}(64,\cdot)\) \(\chi_{283}(66,\cdot)\) \(\chi_{283}(71,\cdot)\) \(\chi_{283}(78,\cdot)\) \(\chi_{283}(86,\cdot)\) \(\chi_{283}(106,\cdot)\) \(\chi_{283}(111,\cdot)\) \(\chi_{283}(116,\cdot)\) \(\chi_{283}(127,\cdot)\) \(\chi_{283}(134,\cdot)\) \(\chi_{283}(141,\cdot)\) \(\chi_{283}(151,\cdot)\) \(\chi_{283}(152,\cdot)\) \(\chi_{283}(155,\cdot)\) \(\chi_{283}(158,\cdot)\) \(\chi_{283}(161,\cdot)\) \(\chi_{283}(163,\cdot)\) \(\chi_{283}(168,\cdot)\) \(\chi_{283}(175,\cdot)\) \(\chi_{283}(181,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{36}{47}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{37}{47}\right)\)\(e\left(\frac{36}{47}\right)\)\(e\left(\frac{27}{47}\right)\)\(e\left(\frac{22}{47}\right)\)\(e\left(\frac{26}{47}\right)\)\(e\left(\frac{44}{47}\right)\)\(e\left(\frac{17}{47}\right)\)\(e\left(\frac{25}{47}\right)\)\(e\left(\frac{12}{47}\right)\)\(e\left(\frac{20}{47}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{47})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 283 }(134,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{283}(134,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(134,r) e\left(\frac{2r}{283}\right) = -16.5118464686+3.2185285767i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 283 }(134,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{283}(134,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(134,r) \chi_{283}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 283 }(134,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{283}(134,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(134,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = 14.103852515+-11.137701168i \)