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Properties

Label 6241.44
Modulus $6241$
Conductor $6241$
Order $3081$
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(6241, base_ring=CyclotomicField(6162)) M = H._module chi = DirichletCharacter(H, M([4912]))
 
Copy content gp:[g,chi] = znchar(Mod(44, 6241))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6241.44");
 

Basic properties

Modulus: \(6241\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6241\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3081\)
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 6241.o

\(\chi_{6241}(2,\cdot)\) \(\chi_{6241}(4,\cdot)\) \(\chi_{6241}(5,\cdot)\) \(\chi_{6241}(9,\cdot)\) \(\chi_{6241}(11,\cdot)\) \(\chi_{6241}(13,\cdot)\) \(\chi_{6241}(16,\cdot)\) \(\chi_{6241}(19,\cdot)\) \(\chi_{6241}(20,\cdot)\) \(\chi_{6241}(25,\cdot)\) \(\chi_{6241}(26,\cdot)\) \(\chi_{6241}(32,\cdot)\) \(\chi_{6241}(36,\cdot)\) \(\chi_{6241}(40,\cdot)\) \(\chi_{6241}(42,\cdot)\) \(\chi_{6241}(44,\cdot)\) \(\chi_{6241}(45,\cdot)\) \(\chi_{6241}(49,\cdot)\) \(\chi_{6241}(50,\cdot)\) \(\chi_{6241}(51,\cdot)\) \(\chi_{6241}(72,\cdot)\) \(\chi_{6241}(73,\cdot)\) \(\chi_{6241}(76,\cdot)\) \(\chi_{6241}(81,\cdot)\) \(\chi_{6241}(83,\cdot)\) \(\chi_{6241}(84,\cdot)\) \(\chi_{6241}(88,\cdot)\) \(\chi_{6241}(90,\cdot)\) \(\chi_{6241}(92,\cdot)\) \(\chi_{6241}(95,\cdot)\) ...

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: $\Q(\zeta_{3081})$
Fixed field: Number field defined by a degree 3081 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{2456}{3081}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6241 }(44, a) \) \(1\)\(1\)\(e\left(\frac{2687}{3081}\right)\)\(e\left(\frac{2456}{3081}\right)\)\(e\left(\frac{2293}{3081}\right)\)\(e\left(\frac{445}{3081}\right)\)\(e\left(\frac{2062}{3081}\right)\)\(e\left(\frac{2248}{3081}\right)\)\(e\left(\frac{633}{1027}\right)\)\(e\left(\frac{1831}{3081}\right)\)\(e\left(\frac{17}{1027}\right)\)\(e\left(\frac{2545}{3081}\right)\)
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)
 
\( \chi_{ 6241 }(44,a) \;\) at \(\;a = \) e.g. 2