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Properties

Label 3539.55
Modulus $3539$
Conductor $3539$
Order $1769$
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(3539, base_ring=CyclotomicField(3538)) M = H._module chi = DirichletCharacter(H, M([1304]))
 
Copy content gp:[g,chi] = znchar(Mod(55, 3539))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3539.55");
 

Basic properties

Modulus: \(3539\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3539\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1769\)
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 3539.g

\(\chi_{3539}(3,\cdot)\) \(\chi_{3539}(4,\cdot)\) \(\chi_{3539}(5,\cdot)\) \(\chi_{3539}(9,\cdot)\) \(\chi_{3539}(11,\cdot)\) \(\chi_{3539}(12,\cdot)\) \(\chi_{3539}(14,\cdot)\) \(\chi_{3539}(15,\cdot)\) \(\chi_{3539}(16,\cdot)\) \(\chi_{3539}(20,\cdot)\) \(\chi_{3539}(23,\cdot)\) \(\chi_{3539}(25,\cdot)\) \(\chi_{3539}(27,\cdot)\) \(\chi_{3539}(29,\cdot)\) \(\chi_{3539}(34,\cdot)\) \(\chi_{3539}(36,\cdot)\) \(\chi_{3539}(38,\cdot)\) \(\chi_{3539}(42,\cdot)\) \(\chi_{3539}(44,\cdot)\) \(\chi_{3539}(45,\cdot)\) \(\chi_{3539}(48,\cdot)\) \(\chi_{3539}(49,\cdot)\) \(\chi_{3539}(52,\cdot)\) \(\chi_{3539}(55,\cdot)\) \(\chi_{3539}(56,\cdot)\) \(\chi_{3539}(59,\cdot)\) \(\chi_{3539}(60,\cdot)\) \(\chi_{3539}(61,\cdot)\) \(\chi_{3539}(62,\cdot)\) \(\chi_{3539}(64,\cdot)\) ...

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: $\Q(\zeta_{1769})$
Fixed field: Number field defined by a degree 1769 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{652}{1769}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3539 }(55, a) \) \(1\)\(1\)\(e\left(\frac{652}{1769}\right)\)\(e\left(\frac{140}{1769}\right)\)\(e\left(\frac{1304}{1769}\right)\)\(e\left(\frac{984}{1769}\right)\)\(e\left(\frac{792}{1769}\right)\)\(e\left(\frac{703}{1769}\right)\)\(e\left(\frac{187}{1769}\right)\)\(e\left(\frac{280}{1769}\right)\)\(e\left(\frac{1636}{1769}\right)\)\(e\left(\frac{104}{1769}\right)\)
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)
 
\( \chi_{ 3539 }(55,a) \;\) at \(\;a = \) e.g. 2