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Properties

Label 5987.2
Modulus $5987$
Conductor $5987$
Order $5986$
Real no
Primitive yes
Minimal yes
Parity odd

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(5987, base_ring=CyclotomicField(5986)) M = H._module chi = DirichletCharacter(H, M([1]))
 
Copy content gp:[g,chi] = znchar(Mod(2, 5987))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5987.2");
 

Basic properties

Modulus: \(5987\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5987\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(5986\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 5987.h

\(\chi_{5987}(2,\cdot)\) \(\chi_{5987}(5,\cdot)\) \(\chi_{5987}(6,\cdot)\) \(\chi_{5987}(7,\cdot)\) \(\chi_{5987}(8,\cdot)\) \(\chi_{5987}(11,\cdot)\) \(\chi_{5987}(13,\cdot)\) \(\chi_{5987}(15,\cdot)\) \(\chi_{5987}(17,\cdot)\) \(\chi_{5987}(18,\cdot)\) \(\chi_{5987}(20,\cdot)\) \(\chi_{5987}(21,\cdot)\) \(\chi_{5987}(24,\cdot)\) \(\chi_{5987}(28,\cdot)\) \(\chi_{5987}(31,\cdot)\) \(\chi_{5987}(32,\cdot)\) \(\chi_{5987}(33,\cdot)\) \(\chi_{5987}(38,\cdot)\) \(\chi_{5987}(39,\cdot)\) \(\chi_{5987}(43,\cdot)\) \(\chi_{5987}(44,\cdot)\) \(\chi_{5987}(45,\cdot)\) \(\chi_{5987}(46,\cdot)\) \(\chi_{5987}(47,\cdot)\) \(\chi_{5987}(50,\cdot)\) \(\chi_{5987}(51,\cdot)\) \(\chi_{5987}(52,\cdot)\) \(\chi_{5987}(53,\cdot)\) \(\chi_{5987}(54,\cdot)\) \(\chi_{5987}(58,\cdot)\) ...

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: $\Q(\zeta_{2993})$
Fixed field: Number field defined by a degree 5986 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1}{5986}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5987 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{1}{5986}\right)\)\(e\left(\frac{1348}{2993}\right)\)\(e\left(\frac{1}{2993}\right)\)\(e\left(\frac{413}{5986}\right)\)\(e\left(\frac{2697}{5986}\right)\)\(e\left(\frac{97}{5986}\right)\)\(e\left(\frac{3}{5986}\right)\)\(e\left(\frac{2696}{2993}\right)\)\(e\left(\frac{207}{2993}\right)\)\(e\left(\frac{2331}{5986}\right)\)
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)
 
\( \chi_{ 5987 }(2,a) \;\) at \(\;a = \) e.g. 2