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Properties

Label 4729.82
Modulus $4729$
Conductor $4729$
Order $1576$
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(4729, base_ring=CyclotomicField(1576)) M = H._module chi = DirichletCharacter(H, M([761]))
 
Copy content gp:[g,chi] = znchar(Mod(82, 4729))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4729.82");
 

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: \(1576\)
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 4729.n

\(\chi_{4729}(11,\cdot)\) \(\chi_{4729}(22,\cdot)\) \(\chi_{4729}(23,\cdot)\) \(\chi_{4729}(29,\cdot)\) \(\chi_{4729}(31,\cdot)\) \(\chi_{4729}(33,\cdot)\) \(\chi_{4729}(41,\cdot)\) \(\chi_{4729}(44,\cdot)\) \(\chi_{4729}(46,\cdot)\) \(\chi_{4729}(62,\cdot)\) \(\chi_{4729}(66,\cdot)\) \(\chi_{4729}(69,\cdot)\) \(\chi_{4729}(82,\cdot)\) \(\chi_{4729}(87,\cdot)\) \(\chi_{4729}(88,\cdot)\) \(\chi_{4729}(92,\cdot)\) \(\chi_{4729}(93,\cdot)\) \(\chi_{4729}(99,\cdot)\) \(\chi_{4729}(103,\cdot)\) \(\chi_{4729}(113,\cdot)\) \(\chi_{4729}(116,\cdot)\) \(\chi_{4729}(123,\cdot)\) \(\chi_{4729}(124,\cdot)\) \(\chi_{4729}(132,\cdot)\) \(\chi_{4729}(138,\cdot)\) \(\chi_{4729}(151,\cdot)\) \(\chi_{4729}(163,\cdot)\) \(\chi_{4729}(164,\cdot)\) \(\chi_{4729}(173,\cdot)\) \(\chi_{4729}(174,\cdot)\) ...

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: $\Q(\zeta_{1576})$
Fixed field: Number field defined by a degree 1576 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{761}{1576}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(82, a) \) \(-1\)\(1\)\(e\left(\frac{435}{788}\right)\)\(e\left(\frac{463}{788}\right)\)\(e\left(\frac{41}{394}\right)\)\(e\left(\frac{595}{788}\right)\)\(e\left(\frac{55}{394}\right)\)\(e\left(\frac{261}{788}\right)\)\(e\left(\frac{517}{788}\right)\)\(e\left(\frac{69}{394}\right)\)\(e\left(\frac{121}{394}\right)\)\(e\left(\frac{1295}{1576}\right)\)
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)
 
\( \chi_{ 4729 }(82,a) \;\) at \(\;a = \) e.g. 2