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Properties

Label 4999.78
Modulus $4999$
Conductor $4999$
Order $2499$
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(4999, base_ring=CyclotomicField(4998)) M = H._module chi = DirichletCharacter(H, M([4264]))
 
Copy content gp:[g,chi] = znchar(Mod(78, 4999))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4999.78");
 

Basic properties

Modulus: \(4999\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4999\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2499\)
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 4999.w

\(\chi_{4999}(9,\cdot)\) \(\chi_{4999}(17,\cdot)\) \(\chi_{4999}(18,\cdot)\) \(\chi_{4999}(19,\cdot)\) \(\chi_{4999}(21,\cdot)\) \(\chi_{4999}(33,\cdot)\) \(\chi_{4999}(37,\cdot)\) \(\chi_{4999}(38,\cdot)\) \(\chi_{4999}(39,\cdot)\) \(\chi_{4999}(42,\cdot)\) \(\chi_{4999}(45,\cdot)\) \(\chi_{4999}(49,\cdot)\) \(\chi_{4999}(53,\cdot)\) \(\chi_{4999}(59,\cdot)\) \(\chi_{4999}(66,\cdot)\) \(\chi_{4999}(67,\cdot)\) \(\chi_{4999}(68,\cdot)\) \(\chi_{4999}(69,\cdot)\) \(\chi_{4999}(72,\cdot)\) \(\chi_{4999}(73,\cdot)\) \(\chi_{4999}(77,\cdot)\) \(\chi_{4999}(78,\cdot)\) \(\chi_{4999}(81,\cdot)\) \(\chi_{4999}(85,\cdot)\) \(\chi_{4999}(90,\cdot)\) \(\chi_{4999}(91,\cdot)\) \(\chi_{4999}(93,\cdot)\) \(\chi_{4999}(95,\cdot)\) \(\chi_{4999}(98,\cdot)\) \(\chi_{4999}(105,\cdot)\) ...

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: $\Q(\zeta_{2499})$
Fixed field: Number field defined by a degree 2499 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{2132}{2499}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4999 }(78, a) \) \(1\)\(1\)\(e\left(\frac{181}{357}\right)\)\(e\left(\frac{2132}{2499}\right)\)\(e\left(\frac{5}{357}\right)\)\(e\left(\frac{44}{119}\right)\)\(e\left(\frac{300}{833}\right)\)\(e\left(\frac{2090}{2499}\right)\)\(e\left(\frac{62}{119}\right)\)\(e\left(\frac{1765}{2499}\right)\)\(e\left(\frac{313}{357}\right)\)\(e\left(\frac{257}{357}\right)\)
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)
 
\( \chi_{ 4999 }(78,a) \;\) at \(\;a = \) e.g. 2