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Properties

Label 4327.1074
Modulus $4327$
Conductor $4327$
Order $721$
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(4327, base_ring=CyclotomicField(1442)) M = H._module chi = DirichletCharacter(H, M([1128]))
 
Copy content gp:[g,chi] = znchar(Mod(1074, 4327))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4327.1074");
 

Basic properties

Modulus: \(4327\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4327\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(721\)
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 4327.m

\(\chi_{4327}(8,\cdot)\) \(\chi_{4327}(18,\cdot)\) \(\chi_{4327}(19,\cdot)\) \(\chi_{4327}(25,\cdot)\) \(\chi_{4327}(55,\cdot)\) \(\chi_{4327}(60,\cdot)\) \(\chi_{4327}(64,\cdot)\) \(\chi_{4327}(65,\cdot)\) \(\chi_{4327}(68,\cdot)\) \(\chi_{4327}(70,\cdot)\) \(\chi_{4327}(93,\cdot)\) \(\chi_{4327}(101,\cdot)\) \(\chi_{4327}(111,\cdot)\) \(\chi_{4327}(115,\cdot)\) \(\chi_{4327}(116,\cdot)\) \(\chi_{4327}(121,\cdot)\) \(\chi_{4327}(132,\cdot)\) \(\chi_{4327}(135,\cdot)\) \(\chi_{4327}(141,\cdot)\) \(\chi_{4327}(143,\cdot)\) \(\chi_{4327}(144,\cdot)\) \(\chi_{4327}(152,\cdot)\) \(\chi_{4327}(153,\cdot)\) \(\chi_{4327}(154,\cdot)\) \(\chi_{4327}(168,\cdot)\) \(\chi_{4327}(169,\cdot)\) \(\chi_{4327}(182,\cdot)\) \(\chi_{4327}(193,\cdot)\) \(\chi_{4327}(200,\cdot)\) \(\chi_{4327}(227,\cdot)\) ...

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: $\Q(\zeta_{721})$
Fixed field: Number field defined by a degree 721 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{564}{721}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4327 }(1074, a) \) \(1\)\(1\)\(e\left(\frac{116}{721}\right)\)\(e\left(\frac{564}{721}\right)\)\(e\left(\frac{232}{721}\right)\)\(e\left(\frac{328}{721}\right)\)\(e\left(\frac{680}{721}\right)\)\(e\left(\frac{437}{721}\right)\)\(e\left(\frac{348}{721}\right)\)\(e\left(\frac{407}{721}\right)\)\(e\left(\frac{444}{721}\right)\)\(e\left(\frac{391}{721}\right)\)
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)
 
\( \chi_{ 4327 }(1074,a) \;\) at \(\;a = \) e.g. 2