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Properties

Label 4451.59
Modulus $4451$
Conductor $4451$
Order $890$
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(4451, base_ring=CyclotomicField(890)) M = H._module chi = DirichletCharacter(H, M([321]))
 
Copy content gp:[g,chi] = znchar(Mod(59, 4451))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4451.59");
 

Basic properties

Modulus: \(4451\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4451\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(890\)
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 4451.j

\(\chi_{4451}(19,\cdot)\) \(\chi_{4451}(22,\cdot)\) \(\chi_{4451}(23,\cdot)\) \(\chi_{4451}(30,\cdot)\) \(\chi_{4451}(32,\cdot)\) \(\chi_{4451}(51,\cdot)\) \(\chi_{4451}(59,\cdot)\) \(\chi_{4451}(65,\cdot)\) \(\chi_{4451}(82,\cdot)\) \(\chi_{4451}(117,\cdot)\) \(\chi_{4451}(119,\cdot)\) \(\chi_{4451}(126,\cdot)\) \(\chi_{4451}(155,\cdot)\) \(\chi_{4451}(166,\cdot)\) \(\chi_{4451}(200,\cdot)\) \(\chi_{4451}(226,\cdot)\) \(\chi_{4451}(268,\cdot)\) \(\chi_{4451}(269,\cdot)\) \(\chi_{4451}(276,\cdot)\) \(\chi_{4451}(294,\cdot)\) \(\chi_{4451}(303,\cdot)\) \(\chi_{4451}(317,\cdot)\) \(\chi_{4451}(319,\cdot)\) \(\chi_{4451}(321,\cdot)\) \(\chi_{4451}(326,\cdot)\) \(\chi_{4451}(340,\cdot)\) \(\chi_{4451}(353,\cdot)\) \(\chi_{4451}(384,\cdot)\) \(\chi_{4451}(419,\cdot)\) \(\chi_{4451}(435,\cdot)\) ...

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: $\Q(\zeta_{445})$
Fixed field: Number field defined by a degree 890 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{321}{890}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4451 }(59, a) \) \(-1\)\(1\)\(e\left(\frac{321}{890}\right)\)\(e\left(\frac{139}{445}\right)\)\(e\left(\frac{321}{445}\right)\)\(e\left(\frac{358}{445}\right)\)\(e\left(\frac{599}{890}\right)\)\(e\left(\frac{169}{445}\right)\)\(e\left(\frac{73}{890}\right)\)\(e\left(\frac{278}{445}\right)\)\(e\left(\frac{147}{890}\right)\)\(e\left(\frac{297}{445}\right)\)
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)
 
\( \chi_{ 4451 }(59,a) \;\) at \(\;a = \) e.g. 2