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Properties

Label 4508.487
Modulus $4508$
Conductor $4508$
Order $462$
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(4508, base_ring=CyclotomicField(462)) M = H._module chi = DirichletCharacter(H, M([231,242,84]))
 
Copy content gp:[g,chi] = znchar(Mod(487, 4508))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4508.487");
 

Basic properties

Modulus: \(4508\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4508\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(462\)
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 4508.cj

\(\chi_{4508}(39,\cdot)\) \(\chi_{4508}(95,\cdot)\) \(\chi_{4508}(123,\cdot)\) \(\chi_{4508}(151,\cdot)\) \(\chi_{4508}(163,\cdot)\) \(\chi_{4508}(179,\cdot)\) \(\chi_{4508}(219,\cdot)\) \(\chi_{4508}(303,\cdot)\) \(\chi_{4508}(331,\cdot)\) \(\chi_{4508}(347,\cdot)\) \(\chi_{4508}(403,\cdot)\) \(\chi_{4508}(443,\cdot)\) \(\chi_{4508}(487,\cdot)\) \(\chi_{4508}(499,\cdot)\) \(\chi_{4508}(515,\cdot)\) \(\chi_{4508}(555,\cdot)\) \(\chi_{4508}(583,\cdot)\) \(\chi_{4508}(611,\cdot)\) \(\chi_{4508}(627,\cdot)\) \(\chi_{4508}(639,\cdot)\) \(\chi_{4508}(683,\cdot)\) \(\chi_{4508}(739,\cdot)\) \(\chi_{4508}(767,\cdot)\) \(\chi_{4508}(795,\cdot)\) \(\chi_{4508}(807,\cdot)\) \(\chi_{4508}(823,\cdot)\) \(\chi_{4508}(947,\cdot)\) \(\chi_{4508}(975,\cdot)\) \(\chi_{4508}(991,\cdot)\) \(\chi_{4508}(1087,\cdot)\) ...

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: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

Values on generators

\((2255,1473,1569)\) → \((-1,e\left(\frac{11}{21}\right),e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(25\)\(27\)
\( \chi_{ 4508 }(487, a) \) \(-1\)\(1\)\(e\left(\frac{431}{462}\right)\)\(e\left(\frac{86}{231}\right)\)\(e\left(\frac{200}{231}\right)\)\(e\left(\frac{41}{462}\right)\)\(e\left(\frac{64}{77}\right)\)\(e\left(\frac{47}{154}\right)\)\(e\left(\frac{85}{231}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{172}{231}\right)\)\(e\left(\frac{123}{154}\right)\)
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)
 
\( \chi_{ 4508 }(487,a) \;\) at \(\;a = \) e.g. 2