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Properties

Label 2683.1024
Modulus $2683$
Conductor $2683$
Order $1341$
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(2683, base_ring=CyclotomicField(2682)) M = H._module chi = DirichletCharacter(H, M([10]))
 
Copy content gp:[g,chi] = znchar(Mod(1024, 2683))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2683.1024");
 

Basic properties

Modulus: \(2683\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2683\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1341\)
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 2683.k

\(\chi_{2683}(4,\cdot)\) \(\chi_{2683}(6,\cdot)\) \(\chi_{2683}(9,\cdot)\) \(\chi_{2683}(11,\cdot)\) \(\chi_{2683}(14,\cdot)\) \(\chi_{2683}(16,\cdot)\) \(\chi_{2683}(21,\cdot)\) \(\chi_{2683}(23,\cdot)\) \(\chi_{2683}(24,\cdot)\) \(\chi_{2683}(25,\cdot)\) \(\chi_{2683}(26,\cdot)\) \(\chi_{2683}(34,\cdot)\) \(\chi_{2683}(35,\cdot)\) \(\chi_{2683}(36,\cdot)\) \(\chi_{2683}(38,\cdot)\) \(\chi_{2683}(39,\cdot)\) \(\chi_{2683}(40,\cdot)\) \(\chi_{2683}(41,\cdot)\) \(\chi_{2683}(51,\cdot)\) \(\chi_{2683}(54,\cdot)\) \(\chi_{2683}(57,\cdot)\) \(\chi_{2683}(58,\cdot)\) \(\chi_{2683}(60,\cdot)\) \(\chi_{2683}(65,\cdot)\) \(\chi_{2683}(67,\cdot)\) \(\chi_{2683}(71,\cdot)\) \(\chi_{2683}(73,\cdot)\) \(\chi_{2683}(74,\cdot)\) \(\chi_{2683}(81,\cdot)\) \(\chi_{2683}(85,\cdot)\) ...

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: $\Q(\zeta_{1341})$
Fixed field: Number field defined by a degree 1341 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{5}{1341}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2683 }(1024, a) \) \(1\)\(1\)\(e\left(\frac{5}{1341}\right)\)\(e\left(\frac{557}{1341}\right)\)\(e\left(\frac{10}{1341}\right)\)\(e\left(\frac{127}{1341}\right)\)\(e\left(\frac{562}{1341}\right)\)\(e\left(\frac{292}{447}\right)\)\(e\left(\frac{5}{447}\right)\)\(e\left(\frac{1114}{1341}\right)\)\(e\left(\frac{44}{447}\right)\)\(e\left(\frac{29}{1341}\right)\)
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)
 
\( \chi_{ 2683 }(1024,a) \;\) at \(\;a = \) e.g. 2