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Properties

Label 4987.73
Modulus $4987$
Conductor $4987$
Order $2493$
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(4987, base_ring=CyclotomicField(4986)) M = H._module chi = DirichletCharacter(H, M([4582]))
 
Copy content gp:[g,chi] = znchar(Mod(73, 4987))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4987.73");
 

Basic properties

Modulus: \(4987\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4987\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2493\)
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 4987.k

\(\chi_{4987}(4,\cdot)\) \(\chi_{4987}(7,\cdot)\) \(\chi_{4987}(9,\cdot)\) \(\chi_{4987}(10,\cdot)\) \(\chi_{4987}(16,\cdot)\) \(\chi_{4987}(22,\cdot)\) \(\chi_{4987}(25,\cdot)\) \(\chi_{4987}(26,\cdot)\) \(\chi_{4987}(28,\cdot)\) \(\chi_{4987}(34,\cdot)\) \(\chi_{4987}(38,\cdot)\) \(\chi_{4987}(40,\cdot)\) \(\chi_{4987}(42,\cdot)\) \(\chi_{4987}(43,\cdot)\) \(\chi_{4987}(47,\cdot)\) \(\chi_{4987}(49,\cdot)\) \(\chi_{4987}(51,\cdot)\) \(\chi_{4987}(54,\cdot)\) \(\chi_{4987}(55,\cdot)\) \(\chi_{4987}(57,\cdot)\) \(\chi_{4987}(60,\cdot)\) \(\chi_{4987}(61,\cdot)\) \(\chi_{4987}(65,\cdot)\) \(\chi_{4987}(70,\cdot)\) \(\chi_{4987}(71,\cdot)\) \(\chi_{4987}(73,\cdot)\) \(\chi_{4987}(81,\cdot)\) \(\chi_{4987}(82,\cdot)\) \(\chi_{4987}(85,\cdot)\) \(\chi_{4987}(88,\cdot)\) ...

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: $\Q(\zeta_{2493})$
Fixed field: Number field defined by a degree 2493 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{2291}{2493}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4987 }(73, a) \) \(1\)\(1\)\(e\left(\frac{2291}{2493}\right)\)\(e\left(\frac{52}{2493}\right)\)\(e\left(\frac{2089}{2493}\right)\)\(e\left(\frac{656}{2493}\right)\)\(e\left(\frac{781}{831}\right)\)\(e\left(\frac{2353}{2493}\right)\)\(e\left(\frac{629}{831}\right)\)\(e\left(\frac{104}{2493}\right)\)\(e\left(\frac{454}{2493}\right)\)\(e\left(\frac{2354}{2493}\right)\)
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)
 
\( \chi_{ 4987 }(73,a) \;\) at \(\;a = \) e.g. 2