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Properties

Label 4501.618
Modulus $4501$
Conductor $4501$
Order $642$
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(4501, base_ring=CyclotomicField(642)) M = H._module chi = DirichletCharacter(H, M([214,375]))
 
Copy content gp:[g,chi] = znchar(Mod(618, 4501))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4501.618");
 

Basic properties

Modulus: \(4501\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4501\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(642\)
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 4501.be

\(\chi_{4501}(2,\cdot)\) \(\chi_{4501}(18,\cdot)\) \(\chi_{4501}(30,\cdot)\) \(\chi_{4501}(32,\cdot)\) \(\chi_{4501}(67,\cdot)\) \(\chi_{4501}(72,\cdot)\) \(\chi_{4501}(107,\cdot)\) \(\chi_{4501}(128,\cdot)\) \(\chi_{4501}(172,\cdot)\) \(\chi_{4501}(200,\cdot)\) \(\chi_{4501}(268,\cdot)\) \(\chi_{4501}(270,\cdot)\) \(\chi_{4501}(284,\cdot)\) \(\chi_{4501}(319,\cdot)\) \(\chi_{4501}(326,\cdot)\) \(\chi_{4501}(387,\cdot)\) \(\chi_{4501}(389,\cdot)\) \(\chi_{4501}(403,\cdot)\) \(\chi_{4501}(422,\cdot)\) \(\chi_{4501}(429,\cdot)\) \(\chi_{4501}(450,\cdot)\) \(\chi_{4501}(480,\cdot)\) \(\chi_{4501}(499,\cdot)\) \(\chi_{4501}(501,\cdot)\) \(\chi_{4501}(508,\cdot)\) \(\chi_{4501}(543,\cdot)\) \(\chi_{4501}(557,\cdot)\) \(\chi_{4501}(562,\cdot)\) \(\chi_{4501}(583,\cdot)\) \(\chi_{4501}(618,\cdot)\) ...

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: $\Q(\zeta_{321})$
Fixed field: Number field defined by a degree 642 polynomial (not computed)

Values on generators

\((3216,1940)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{125}{214}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 4501 }(618, a) \) \(-1\)\(1\)\(e\left(\frac{173}{642}\right)\)\(e\left(\frac{529}{642}\right)\)\(e\left(\frac{173}{321}\right)\)\(e\left(\frac{281}{642}\right)\)\(e\left(\frac{10}{107}\right)\)\(e\left(\frac{173}{214}\right)\)\(e\left(\frac{208}{321}\right)\)\(e\left(\frac{227}{321}\right)\)\(e\left(\frac{589}{642}\right)\)\(e\left(\frac{233}{642}\right)\)
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)
 
\( \chi_{ 4501 }(618,a) \;\) at \(\;a = \) e.g. 2