Properties

Label 283.23
Modulus $283$
Conductor $283$
Order $141$
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(282))
 
M = H._module
 
chi = DirichletCharacter(H, M([100]))
 
pari: [g,chi] = znchar(Mod(23,283))
 

Basic properties

Modulus: \(283\)
Conductor: \(283\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(141\)
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.g

\(\chi_{283}(6,\cdot)\) \(\chi_{283}(7,\cdot)\) \(\chi_{283}(9,\cdot)\) \(\chi_{283}(10,\cdot)\) \(\chi_{283}(11,\cdot)\) \(\chi_{283}(13,\cdot)\) \(\chi_{283}(23,\cdot)\) \(\chi_{283}(24,\cdot)\) \(\chi_{283}(25,\cdot)\) \(\chi_{283}(28,\cdot)\) \(\chi_{283}(34,\cdot)\) \(\chi_{283}(36,\cdot)\) \(\chi_{283}(40,\cdot)\) \(\chi_{283}(41,\cdot)\) \(\chi_{283}(49,\cdot)\) \(\chi_{283}(52,\cdot)\) \(\chi_{283}(57,\cdot)\) \(\chi_{283}(59,\cdot)\) \(\chi_{283}(62,\cdot)\) \(\chi_{283}(63,\cdot)\) \(\chi_{283}(70,\cdot)\) \(\chi_{283}(73,\cdot)\) \(\chi_{283}(74,\cdot)\) \(\chi_{283}(77,\cdot)\) \(\chi_{283}(81,\cdot)\) \(\chi_{283}(83,\cdot)\) \(\chi_{283}(85,\cdot)\) \(\chi_{283}(89,\cdot)\) \(\chi_{283}(90,\cdot)\) \(\chi_{283}(91,\cdot)\) ...

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: $\Q(\zeta_{141})$
Fixed field: Number field defined by a degree 141 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{50}{141}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 283 }(23, a) \) \(1\)\(1\)\(e\left(\frac{18}{47}\right)\)\(e\left(\frac{50}{141}\right)\)\(e\left(\frac{36}{47}\right)\)\(e\left(\frac{88}{141}\right)\)\(e\left(\frac{104}{141}\right)\)\(e\left(\frac{82}{141}\right)\)\(e\left(\frac{7}{47}\right)\)\(e\left(\frac{100}{141}\right)\)\(e\left(\frac{1}{141}\right)\)\(e\left(\frac{127}{141}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 283 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(23,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2