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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup({modulus}, base_ring=CyclotomicField({sage_zeta_order})) M = H._module chi = DirichletCharacter(H, M([{sage_dirichlet_gens}]))
 
Copy content gp:[g,chi] = znchar(Mod({number}, {modulus}))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("{modulus}.{number}");
 

Kronecker symbol representation

Copy content comment:Kronecker symbol
 
Copy content sage:kronecker_character({symbol_num})
 
Copy content gp:znchartokronecker(g,chi)
 
Copy content magma:KroneckerCharacter({symbol_num});
 

\(\displaystyle\left(\frac{-4297259}{\bullet}\right)\)

Basic properties

Modulus: \(4297259\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4297259\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: yes
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Related number fields

Field of values: \(\Q\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 

Values on generators

\(2\) → \(-1\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)
\( \chi_{ 4297259 }(4297258, a) \) \(-1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(-1\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 4297259 }(4297258,a) \;\) at \(\;a = \) e.g. 2