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Properties

Label 6427.135
Modulus $6427$
Conductor $6427$
Order $1071$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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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(6427, base_ring=CyclotomicField(2142)) M = H._module chi = DirichletCharacter(H, M([806]))
 
Copy content gp:[g,chi] = znchar(Mod(135, 6427))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6427.135");
 

Basic properties

Modulus: \(6427\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6427\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1071\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
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);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 6427.bc

\(\chi_{6427}(4,\cdot)\) \(\chi_{6427}(10,\cdot)\) \(\chi_{6427}(16,\cdot)\) \(\chi_{6427}(23,\cdot)\) \(\chi_{6427}(33,\cdot)\) \(\chi_{6427}(40,\cdot)\) \(\chi_{6427}(54,\cdot)\) \(\chi_{6427}(57,\cdot)\) \(\chi_{6427}(87,\cdot)\) \(\chi_{6427}(97,\cdot)\) \(\chi_{6427}(100,\cdot)\) \(\chi_{6427}(132,\cdot)\) \(\chi_{6427}(135,\cdot)\) \(\chi_{6427}(139,\cdot)\) \(\chi_{6427}(142,\cdot)\) \(\chi_{6427}(153,\cdot)\) \(\chi_{6427}(156,\cdot)\) \(\chi_{6427}(178,\cdot)\) \(\chi_{6427}(201,\cdot)\) \(\chi_{6427}(206,\cdot)\) \(\chi_{6427}(210,\cdot)\) \(\chi_{6427}(216,\cdot)\) \(\chi_{6427}(222,\cdot)\) \(\chi_{6427}(230,\cdot)\) \(\chi_{6427}(238,\cdot)\) \(\chi_{6427}(241,\cdot)\) \(\chi_{6427}(250,\cdot)\) \(\chi_{6427}(256,\cdot)\) \(\chi_{6427}(257,\cdot)\) \(\chi_{6427}(271,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{1071})$
Fixed field: Number field defined by a degree 1071 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{403}{1071}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6427 }(135, a) \) \(1\)\(1\)\(e\left(\frac{298}{357}\right)\)\(e\left(\frac{403}{1071}\right)\)\(e\left(\frac{239}{357}\right)\)\(e\left(\frac{37}{51}\right)\)\(e\left(\frac{226}{1071}\right)\)\(e\left(\frac{8}{153}\right)\)\(e\left(\frac{60}{119}\right)\)\(e\left(\frac{806}{1071}\right)\)\(e\left(\frac{200}{357}\right)\)\(e\left(\frac{10}{63}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 6427 }(135,a) \;\) at \(\;a = \) e.g. 2