Properties

Label 283.21
Modulus $283$
Conductor $283$
Order $94$
Real no
Primitive yes
Minimal yes
Parity odd

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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([93]))
 
pari: [g,chi] = znchar(Mod(21,283))
 

Basic properties

Modulus: \(283\)
Conductor: \(283\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(94\)
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 283.f

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

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 94 polynomial

Values on generators

\(3\) → \(e\left(\frac{93}{94}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 283 }(21, a) \) \(-1\)\(1\)\(e\left(\frac{29}{94}\right)\)\(e\left(\frac{93}{94}\right)\)\(e\left(\frac{29}{47}\right)\)\(e\left(\frac{49}{94}\right)\)\(e\left(\frac{14}{47}\right)\)\(e\left(\frac{2}{47}\right)\)\(e\left(\frac{87}{94}\right)\)\(e\left(\frac{46}{47}\right)\)\(e\left(\frac{39}{47}\right)\)\(e\left(\frac{18}{47}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 283 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 283 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 283 }(21,·),\chi_{ 283 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 283 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2