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Properties

Label 1973.1025
Modulus $1973$
Conductor $1973$
Order $493$
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(1973, base_ring=CyclotomicField(986)) M = H._module chi = DirichletCharacter(H, M([384]))
 
Copy content gp:[g,chi] = znchar(Mod(1025, 1973))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1973.1025");
 

Basic properties

Modulus: \(1973\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1973\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(493\)
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 1973.j

\(\chi_{1973}(14,\cdot)\) \(\chi_{1973}(16,\cdot)\) \(\chi_{1973}(21,\cdot)\) \(\chi_{1973}(31,\cdot)\) \(\chi_{1973}(35,\cdot)\) \(\chi_{1973}(36,\cdot)\) \(\chi_{1973}(40,\cdot)\) \(\chi_{1973}(44,\cdot)\) \(\chi_{1973}(54,\cdot)\) \(\chi_{1973}(59,\cdot)\) \(\chi_{1973}(60,\cdot)\) \(\chi_{1973}(66,\cdot)\) \(\chi_{1973}(73,\cdot)\) \(\chi_{1973}(76,\cdot)\) \(\chi_{1973}(81,\cdot)\) \(\chi_{1973}(90,\cdot)\) \(\chi_{1973}(92,\cdot)\) \(\chi_{1973}(94,\cdot)\) \(\chi_{1973}(99,\cdot)\) \(\chi_{1973}(100,\cdot)\) \(\chi_{1973}(101,\cdot)\) \(\chi_{1973}(102,\cdot)\) \(\chi_{1973}(103,\cdot)\) \(\chi_{1973}(106,\cdot)\) \(\chi_{1973}(114,\cdot)\) \(\chi_{1973}(116,\cdot)\) \(\chi_{1973}(121,\cdot)\) \(\chi_{1973}(127,\cdot)\) \(\chi_{1973}(134,\cdot)\) \(\chi_{1973}(135,\cdot)\) ...

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: $\Q(\zeta_{493})$
Fixed field: Number field defined by a degree 493 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{192}{493}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1973 }(1025, a) \) \(1\)\(1\)\(e\left(\frac{192}{493}\right)\)\(e\left(\frac{53}{493}\right)\)\(e\left(\frac{384}{493}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{245}{493}\right)\)\(e\left(\frac{264}{493}\right)\)\(e\left(\frac{83}{493}\right)\)\(e\left(\frac{106}{493}\right)\)\(e\left(\frac{482}{493}\right)\)\(e\left(\frac{470}{493}\right)\)
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)
 
\( \chi_{ 1973 }(1025,a) \;\) at \(\;a = \) e.g. 2