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Properties

Label 4729.1078
Modulus $4729$
Conductor $4729$
Order $4728$
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(4728)) M = H._module chi = DirichletCharacter(H, M([2537]))
 
Copy content gp:[g,chi] = znchar(Mod(1078, 4729))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4729.1078");
 

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: \(4728\)
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.p

\(\chi_{4729}(17,\cdot)\) \(\chi_{4729}(34,\cdot)\) \(\chi_{4729}(43,\cdot)\) \(\chi_{4729}(47,\cdot)\) \(\chi_{4729}(51,\cdot)\) \(\chi_{4729}(53,\cdot)\) \(\chi_{4729}(55,\cdot)\) \(\chi_{4729}(61,\cdot)\) \(\chi_{4729}(68,\cdot)\) \(\chi_{4729}(77,\cdot)\) \(\chi_{4729}(85,\cdot)\) \(\chi_{4729}(86,\cdot)\) \(\chi_{4729}(89,\cdot)\) \(\chi_{4729}(94,\cdot)\) \(\chi_{4729}(101,\cdot)\) \(\chi_{4729}(102,\cdot)\) \(\chi_{4729}(106,\cdot)\) \(\chi_{4729}(107,\cdot)\) \(\chi_{4729}(109,\cdot)\) \(\chi_{4729}(110,\cdot)\) \(\chi_{4729}(115,\cdot)\) \(\chi_{4729}(119,\cdot)\) \(\chi_{4729}(122,\cdot)\) \(\chi_{4729}(129,\cdot)\) \(\chi_{4729}(136,\cdot)\) \(\chi_{4729}(139,\cdot)\) \(\chi_{4729}(141,\cdot)\) \(\chi_{4729}(143,\cdot)\) \(\chi_{4729}(145,\cdot)\) \(\chi_{4729}(153,\cdot)\) ...

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: $\Q(\zeta_{4728})$
Fixed field: Number field defined by a degree 4728 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{2537}{4728}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(1078, a) \) \(-1\)\(1\)\(e\left(\frac{501}{788}\right)\)\(e\left(\frac{305}{788}\right)\)\(e\left(\frac{107}{394}\right)\)\(e\left(\frac{2083}{2364}\right)\)\(e\left(\frac{9}{394}\right)\)\(e\left(\frac{2005}{2364}\right)\)\(e\left(\frac{715}{788}\right)\)\(e\left(\frac{305}{394}\right)\)\(e\left(\frac{611}{1182}\right)\)\(e\left(\frac{1573}{1576}\right)\)
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)
 
\( \chi_{ 4729 }(1078,a) \;\) at \(\;a = \) e.g. 2