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Properties

Label 3871.748
Modulus $3871$
Conductor $3871$
Order $546$
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(3871, base_ring=CyclotomicField(546)) M = H._module chi = DirichletCharacter(H, M([429,133]))
 
Copy content gp:[g,chi] = znchar(Mod(748, 3871))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3871.748");
 

Basic properties

Modulus: \(3871\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3871\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(546\)
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 3871.cs

\(\chi_{3871}(6,\cdot)\) \(\chi_{3871}(34,\cdot)\) \(\chi_{3871}(118,\cdot)\) \(\chi_{3871}(132,\cdot)\) \(\chi_{3871}(139,\cdot)\) \(\chi_{3871}(153,\cdot)\) \(\chi_{3871}(188,\cdot)\) \(\chi_{3871}(265,\cdot)\) \(\chi_{3871}(272,\cdot)\) \(\chi_{3871}(300,\cdot)\) \(\chi_{3871}(307,\cdot)\) \(\chi_{3871}(314,\cdot)\) \(\chi_{3871}(363,\cdot)\) \(\chi_{3871}(370,\cdot)\) \(\chi_{3871}(384,\cdot)\) \(\chi_{3871}(398,\cdot)\) \(\chi_{3871}(454,\cdot)\) \(\chi_{3871}(461,\cdot)\) \(\chi_{3871}(503,\cdot)\) \(\chi_{3871}(517,\cdot)\) \(\chi_{3871}(559,\cdot)\) \(\chi_{3871}(601,\cdot)\) \(\chi_{3871}(671,\cdot)\) \(\chi_{3871}(692,\cdot)\) \(\chi_{3871}(706,\cdot)\) \(\chi_{3871}(741,\cdot)\) \(\chi_{3871}(748,\cdot)\) \(\chi_{3871}(797,\cdot)\) \(\chi_{3871}(818,\cdot)\) \(\chi_{3871}(825,\cdot)\) ...

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: $\Q(\zeta_{273})$
Fixed field: Number field defined by a degree 546 polynomial (not computed)

Values on generators

\((2845,1030)\) → \((e\left(\frac{11}{14}\right),e\left(\frac{19}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 3871 }(748, a) \) \(1\)\(1\)\(e\left(\frac{110}{273}\right)\)\(e\left(\frac{8}{273}\right)\)\(e\left(\frac{220}{273}\right)\)\(e\left(\frac{485}{546}\right)\)\(e\left(\frac{118}{273}\right)\)\(e\left(\frac{19}{91}\right)\)\(e\left(\frac{16}{273}\right)\)\(e\left(\frac{53}{182}\right)\)\(e\left(\frac{271}{273}\right)\)\(e\left(\frac{76}{91}\right)\)
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)
 
\( \chi_{ 3871 }(748,a) \;\) at \(\;a = \) e.g. 2