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Properties

Label 4229.1063
Modulus $4229$
Conductor $4229$
Order $4228$
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(4229, base_ring=CyclotomicField(4228)) M = H._module chi = DirichletCharacter(H, M([1761]))
 
Copy content gp:[g,chi] = znchar(Mod(1063, 4229))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4229.1063");
 

Basic properties

Modulus: \(4229\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4229\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4228\)
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 4229.l

\(\chi_{4229}(2,\cdot)\) \(\chi_{4229}(3,\cdot)\) \(\chi_{4229}(8,\cdot)\) \(\chi_{4229}(10,\cdot)\) \(\chi_{4229}(12,\cdot)\) \(\chi_{4229}(14,\cdot)\) \(\chi_{4229}(15,\cdot)\) \(\chi_{4229}(18,\cdot)\) \(\chi_{4229}(21,\cdot)\) \(\chi_{4229}(22,\cdot)\) \(\chi_{4229}(23,\cdot)\) \(\chi_{4229}(26,\cdot)\) \(\chi_{4229}(27,\cdot)\) \(\chi_{4229}(31,\cdot)\) \(\chi_{4229}(32,\cdot)\) \(\chi_{4229}(33,\cdot)\) \(\chi_{4229}(34,\cdot)\) \(\chi_{4229}(38,\cdot)\) \(\chi_{4229}(39,\cdot)\) \(\chi_{4229}(40,\cdot)\) \(\chi_{4229}(41,\cdot)\) \(\chi_{4229}(47,\cdot)\) \(\chi_{4229}(48,\cdot)\) \(\chi_{4229}(50,\cdot)\) \(\chi_{4229}(51,\cdot)\) \(\chi_{4229}(56,\cdot)\) \(\chi_{4229}(57,\cdot)\) \(\chi_{4229}(58,\cdot)\) \(\chi_{4229}(59,\cdot)\) \(\chi_{4229}(60,\cdot)\) ...

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: $\Q(\zeta_{4228})$
Fixed field: Number field defined by a degree 4228 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1761}{4228}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4229 }(1063, a) \) \(-1\)\(1\)\(e\left(\frac{1761}{4228}\right)\)\(e\left(\frac{3637}{4228}\right)\)\(e\left(\frac{1761}{2114}\right)\)\(e\left(\frac{563}{2114}\right)\)\(e\left(\frac{585}{2114}\right)\)\(e\left(\frac{31}{1057}\right)\)\(e\left(\frac{1055}{4228}\right)\)\(e\left(\frac{1523}{2114}\right)\)\(e\left(\frac{2887}{4228}\right)\)\(e\left(\frac{1047}{1057}\right)\)
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)
 
\( \chi_{ 4229 }(1063,a) \;\) at \(\;a = \) e.g. 2