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Properties

Label 4729.10
Modulus $4729$
Conductor $4729$
Order $1182$
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([229]))
 
Copy content gp:[g,chi] = znchar(Mod(10, 4729))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4729.10");
 

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: \(1182\)
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.m

\(\chi_{4729}(10,\cdot)\) \(\chi_{4729}(13,\cdot)\) \(\chi_{4729}(15,\cdot)\) \(\chi_{4729}(25,\cdot)\) \(\chi_{4729}(49,\cdot)\) \(\chi_{4729}(56,\cdot)\) \(\chi_{4729}(76,\cdot)\) \(\chi_{4729}(84,\cdot)\) \(\chi_{4729}(97,\cdot)\) \(\chi_{4729}(114,\cdot)\) \(\chi_{4729}(126,\cdot)\) \(\chi_{4729}(140,\cdot)\) \(\chi_{4729}(142,\cdot)\) \(\chi_{4729}(146,\cdot)\) \(\chi_{4729}(148,\cdot)\) \(\chi_{4729}(149,\cdot)\) \(\chi_{4729}(160,\cdot)\) \(\chi_{4729}(171,\cdot)\) \(\chi_{4729}(179,\cdot)\) \(\chi_{4729}(182,\cdot)\) \(\chi_{4729}(189,\cdot)\) \(\chi_{4729}(190,\cdot)\) \(\chi_{4729}(208,\cdot)\) \(\chi_{4729}(210,\cdot)\) \(\chi_{4729}(213,\cdot)\) \(\chi_{4729}(219,\cdot)\) \(\chi_{4729}(222,\cdot)\) \(\chi_{4729}(240,\cdot)\) \(\chi_{4729}(241,\cdot)\) \(\chi_{4729}(247,\cdot)\) ...

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 1182 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{229}{1182}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(10, a) \) \(1\)\(1\)\(e\left(\frac{120}{197}\right)\)\(e\left(\frac{53}{197}\right)\)\(e\left(\frac{43}{197}\right)\)\(e\left(\frac{506}{591}\right)\)\(e\left(\frac{173}{197}\right)\)\(e\left(\frac{413}{591}\right)\)\(e\left(\frac{163}{197}\right)\)\(e\left(\frac{106}{197}\right)\)\(e\left(\frac{275}{591}\right)\)\(e\left(\frac{201}{394}\right)\)
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)
 
\( \chi_{ 4729 }(10,a) \;\) at \(\;a = \) e.g. 2