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Properties

Label 6877.192
Modulus $6877$
Conductor $6877$
Order $1518$
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(6877, base_ring=CyclotomicField(1518)) M = H._module chi = DirichletCharacter(H, M([1265,612]))
 
Copy content gp:[g,chi] = znchar(Mod(192, 6877))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6877.192");
 

Basic properties

Modulus: \(6877\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6877\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1518\)
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 6877.bs

\(\chi_{6877}(4,\cdot)\) \(\chi_{6877}(36,\cdot)\) \(\chi_{6877}(49,\cdot)\) \(\chi_{6877}(62,\cdot)\) \(\chi_{6877}(75,\cdot)\) \(\chi_{6877}(82,\cdot)\) \(\chi_{6877}(95,\cdot)\) \(\chi_{6877}(101,\cdot)\) \(\chi_{6877}(108,\cdot)\) \(\chi_{6877}(121,\cdot)\) \(\chi_{6877}(127,\cdot)\) \(\chi_{6877}(140,\cdot)\) \(\chi_{6877}(147,\cdot)\) \(\chi_{6877}(173,\cdot)\) \(\chi_{6877}(179,\cdot)\) \(\chi_{6877}(186,\cdot)\) \(\chi_{6877}(192,\cdot)\) \(\chi_{6877}(225,\cdot)\) \(\chi_{6877}(238,\cdot)\) \(\chi_{6877}(257,\cdot)\) \(\chi_{6877}(303,\cdot)\) \(\chi_{6877}(335,\cdot)\) \(\chi_{6877}(348,\cdot)\) \(\chi_{6877}(361,\cdot)\) \(\chi_{6877}(374,\cdot)\) \(\chi_{6877}(381,\cdot)\) \(\chi_{6877}(394,\cdot)\) \(\chi_{6877}(400,\cdot)\) \(\chi_{6877}(407,\cdot)\) \(\chi_{6877}(420,\cdot)\) ...

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: $\Q(\zeta_{759})$
Fixed field: Number field defined by a degree 1518 polynomial (not computed)

Values on generators

\((1588,534)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{102}{253}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6877 }(192, a) \) \(1\)\(1\)\(e\left(\frac{707}{1518}\right)\)\(e\left(\frac{595}{759}\right)\)\(e\left(\frac{707}{759}\right)\)\(e\left(\frac{457}{506}\right)\)\(e\left(\frac{379}{1518}\right)\)\(e\left(\frac{265}{1518}\right)\)\(e\left(\frac{201}{506}\right)\)\(e\left(\frac{431}{759}\right)\)\(e\left(\frac{280}{759}\right)\)\(e\left(\frac{1295}{1518}\right)\)
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)
 
\( \chi_{ 6877 }(192,a) \;\) at \(\;a = \) e.g. 2