Properties

Conductor 283
Order 94
Real No
Primitive Yes
Parity Odd
Orbit Label 283.f

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[53]
pari: [g,chi] = znchar(Mod(53,283))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 283
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 94
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 = Odd
Orbit label = 283.f
Orbit index = 6

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 ]

\(\chi_{283}(2,\cdot)\) \(\chi_{283}(8,\cdot)\) \(\chi_{283}(19,\cdot)\) \(\chi_{283}(21,\cdot)\) \(\chi_{283}(27,\cdot)\) \(\chi_{283}(30,\cdot)\) \(\chi_{283}(32,\cdot)\) \(\chi_{283}(33,\cdot)\) \(\chi_{283}(39,\cdot)\) \(\chi_{283}(43,\cdot)\) \(\chi_{283}(53,\cdot)\) \(\chi_{283}(58,\cdot)\) \(\chi_{283}(67,\cdot)\) \(\chi_{283}(76,\cdot)\) \(\chi_{283}(79,\cdot)\) \(\chi_{283}(84,\cdot)\) \(\chi_{283}(102,\cdot)\) \(\chi_{283}(108,\cdot)\) \(\chi_{283}(115,\cdot)\) \(\chi_{283}(120,\cdot)\) \(\chi_{283}(122,\cdot)\) \(\chi_{283}(125,\cdot)\) \(\chi_{283}(128,\cdot)\) \(\chi_{283}(131,\cdot)\) \(\chi_{283}(132,\cdot)\) \(\chi_{283}(142,\cdot)\) \(\chi_{283}(149,\cdot)\) \(\chi_{283}(156,\cdot)\) \(\chi_{283}(167,\cdot)\) \(\chi_{283}(172,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{23}{94}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{85}{94}\right)\)\(e\left(\frac{23}{94}\right)\)\(e\left(\frac{38}{47}\right)\)\(e\left(\frac{1}{94}\right)\)\(e\left(\frac{7}{47}\right)\)\(e\left(\frac{1}{47}\right)\)\(e\left(\frac{67}{94}\right)\)\(e\left(\frac{23}{47}\right)\)\(e\left(\frac{43}{47}\right)\)\(e\left(\frac{9}{47}\right)\)
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)
\( \tau_{ a }( \chi_{ 283 }(53,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{283}(53,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(53,r) e\left(\frac{2r}{283}\right) = -13.088441922+10.5684761462i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 283 }(53,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{283}(53,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(53,r) \chi_{283}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 283 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{283}(53,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(53,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = 3.0565533313+9.8533996992i \)