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Properties

Label 4019.22
Modulus $4019$
Conductor $4019$
Order $2009$
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(4019, base_ring=CyclotomicField(4018)) M = H._module chi = DirichletCharacter(H, M([262]))
 
Copy content gp:[g,chi] = znchar(Mod(22, 4019))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4019.22");
 

Basic properties

Modulus: \(4019\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4019\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2009\)
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 4019.k

\(\chi_{4019}(4,\cdot)\) \(\chi_{4019}(5,\cdot)\) \(\chi_{4019}(12,\cdot)\) \(\chi_{4019}(14,\cdot)\) \(\chi_{4019}(15,\cdot)\) \(\chi_{4019}(16,\cdot)\) \(\chi_{4019}(19,\cdot)\) \(\chi_{4019}(22,\cdot)\) \(\chi_{4019}(23,\cdot)\) \(\chi_{4019}(25,\cdot)\) \(\chi_{4019}(26,\cdot)\) \(\chi_{4019}(36,\cdot)\) \(\chi_{4019}(43,\cdot)\) \(\chi_{4019}(45,\cdot)\) \(\chi_{4019}(49,\cdot)\) \(\chi_{4019}(53,\cdot)\) \(\chi_{4019}(57,\cdot)\) \(\chi_{4019}(58,\cdot)\) \(\chi_{4019}(60,\cdot)\) \(\chi_{4019}(62,\cdot)\) \(\chi_{4019}(64,\cdot)\) \(\chi_{4019}(66,\cdot)\) \(\chi_{4019}(67,\cdot)\) \(\chi_{4019}(69,\cdot)\) \(\chi_{4019}(70,\cdot)\) \(\chi_{4019}(73,\cdot)\) \(\chi_{4019}(74,\cdot)\) \(\chi_{4019}(75,\cdot)\) \(\chi_{4019}(76,\cdot)\) \(\chi_{4019}(77,\cdot)\) ...

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: $\Q(\zeta_{2009})$
Fixed field: Number field defined by a degree 2009 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{131}{2009}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4019 }(22, a) \) \(1\)\(1\)\(e\left(\frac{131}{2009}\right)\)\(e\left(\frac{41}{49}\right)\)\(e\left(\frac{262}{2009}\right)\)\(e\left(\frac{1817}{2009}\right)\)\(e\left(\frac{1812}{2009}\right)\)\(e\left(\frac{1534}{2009}\right)\)\(e\left(\frac{393}{2009}\right)\)\(e\left(\frac{33}{49}\right)\)\(e\left(\frac{1948}{2009}\right)\)\(e\left(\frac{38}{2009}\right)\)
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)
 
\( \chi_{ 4019 }(22,a) \;\) at \(\;a = \) e.g. 2