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Properties

Label 4729.196
Modulus $4729$
Conductor $4729$
Order $591$
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(4729, base_ring=CyclotomicField(1182)) M = H._module chi = DirichletCharacter(H, M([806]))
 
Copy content gp:[g,chi] = znchar(Mod(196, 4729))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4729.196");
 

Basic properties

Modulus: \(4729\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4729\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(591\)
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 4729.k

\(\chi_{4729}(14,\cdot)\) \(\chi_{4729}(19,\cdot)\) \(\chi_{4729}(21,\cdot)\) \(\chi_{4729}(35,\cdot)\) \(\chi_{4729}(37,\cdot)\) \(\chi_{4729}(40,\cdot)\) \(\chi_{4729}(52,\cdot)\) \(\chi_{4729}(60,\cdot)\) \(\chi_{4729}(67,\cdot)\) \(\chi_{4729}(78,\cdot)\) \(\chi_{4729}(83,\cdot)\) \(\chi_{4729}(90,\cdot)\) \(\chi_{4729}(100,\cdot)\) \(\chi_{4729}(117,\cdot)\) \(\chi_{4729}(118,\cdot)\) \(\chi_{4729}(130,\cdot)\) \(\chi_{4729}(131,\cdot)\) \(\chi_{4729}(135,\cdot)\) \(\chi_{4729}(150,\cdot)\) \(\chi_{4729}(169,\cdot)\) \(\chi_{4729}(177,\cdot)\) \(\chi_{4729}(187,\cdot)\) \(\chi_{4729}(193,\cdot)\) \(\chi_{4729}(195,\cdot)\) \(\chi_{4729}(196,\cdot)\) \(\chi_{4729}(224,\cdot)\) \(\chi_{4729}(225,\cdot)\) \(\chi_{4729}(266,\cdot)\) \(\chi_{4729}(294,\cdot)\) \(\chi_{4729}(304,\cdot)\) ...

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: $\Q(\zeta_{591})$
Fixed field: Number field defined by a degree 591 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{403}{591}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(196, a) \) \(1\)\(1\)\(e\left(\frac{166}{197}\right)\)\(e\left(\frac{116}{197}\right)\)\(e\left(\frac{135}{197}\right)\)\(e\left(\frac{457}{591}\right)\)\(e\left(\frac{85}{197}\right)\)\(e\left(\frac{220}{591}\right)\)\(e\left(\frac{104}{197}\right)\)\(e\left(\frac{35}{197}\right)\)\(e\left(\frac{364}{591}\right)\)\(e\left(\frac{75}{197}\right)\)
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)
 
\( \chi_{ 4729 }(196,a) \;\) at \(\;a = \) e.g. 2