Properties

Label 983.2
Modulus $983$
Conductor $983$
Order $491$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(983, base_ring=CyclotomicField(982))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([674]))
 
pari: [g,chi] = znchar(Mod(2,983))
 

Basic properties

Modulus: \(983\)
Conductor: \(983\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(491\)
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 983.c

\(\chi_{983}(2,\cdot)\) \(\chi_{983}(3,\cdot)\) \(\chi_{983}(4,\cdot)\) \(\chi_{983}(6,\cdot)\) \(\chi_{983}(7,\cdot)\) \(\chi_{983}(8,\cdot)\) \(\chi_{983}(9,\cdot)\) \(\chi_{983}(12,\cdot)\) \(\chi_{983}(14,\cdot)\) \(\chi_{983}(16,\cdot)\) \(\chi_{983}(18,\cdot)\) \(\chi_{983}(19,\cdot)\) \(\chi_{983}(21,\cdot)\) \(\chi_{983}(23,\cdot)\) \(\chi_{983}(24,\cdot)\) \(\chi_{983}(25,\cdot)\) \(\chi_{983}(27,\cdot)\) \(\chi_{983}(28,\cdot)\) \(\chi_{983}(31,\cdot)\) \(\chi_{983}(32,\cdot)\) \(\chi_{983}(36,\cdot)\) \(\chi_{983}(37,\cdot)\) \(\chi_{983}(38,\cdot)\) \(\chi_{983}(41,\cdot)\) \(\chi_{983}(42,\cdot)\) \(\chi_{983}(43,\cdot)\) \(\chi_{983}(46,\cdot)\) \(\chi_{983}(47,\cdot)\) \(\chi_{983}(48,\cdot)\) \(\chi_{983}(49,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{491})$
Fixed field: Number field defined by a degree 491 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{337}{491}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 983 }(2, a) \) \(1\)\(1\)\(e\left(\frac{296}{491}\right)\)\(e\left(\frac{80}{491}\right)\)\(e\left(\frac{101}{491}\right)\)\(e\left(\frac{337}{491}\right)\)\(e\left(\frac{376}{491}\right)\)\(e\left(\frac{312}{491}\right)\)\(e\left(\frac{397}{491}\right)\)\(e\left(\frac{160}{491}\right)\)\(e\left(\frac{142}{491}\right)\)\(e\left(\frac{10}{491}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 983 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 983 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 983 }(2,·),\chi_{ 983 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 983 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2