Properties

Label 283.71
Modulus $283$
Conductor $283$
Order $47$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(283, base_ring=CyclotomicField(94))
 
M = H._module
 
chi = DirichletCharacter(H, M([82]))
 
pari: [g,chi] = znchar(Mod(71,283))
 

Basic properties

Modulus: \(283\)
Conductor: \(283\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(47\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 283.e

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

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

Related number fields

Field of values: $\Q(\zeta_{47})$
Fixed field: Number field defined by a degree 47 polynomial

Values on generators

\(3\) → \(e\left(\frac{41}{47}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 283 }(71, a) \) \(1\)\(1\)\(e\left(\frac{33}{47}\right)\)\(e\left(\frac{41}{47}\right)\)\(e\left(\frac{19}{47}\right)\)\(e\left(\frac{12}{47}\right)\)\(e\left(\frac{27}{47}\right)\)\(e\left(\frac{24}{47}\right)\)\(e\left(\frac{5}{47}\right)\)\(e\left(\frac{35}{47}\right)\)\(e\left(\frac{45}{47}\right)\)\(e\left(\frac{28}{47}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 283 }(71,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(71,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(71,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2