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Properties

Label 4447.1042
Modulus $4447$
Conductor $4447$
Order $2223$
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(4447, base_ring=CyclotomicField(4446)) M = H._module chi = DirichletCharacter(H, M([2980]))
 
Copy content gp:[g,chi] = znchar(Mod(1042, 4447))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4447.1042");
 

Basic properties

Modulus: \(4447\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4447\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2223\)
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 4447.w

\(\chi_{4447}(2,\cdot)\) \(\chi_{4447}(4,\cdot)\) \(\chi_{4447}(9,\cdot)\) \(\chi_{4447}(15,\cdot)\) \(\chi_{4447}(16,\cdot)\) \(\chi_{4447}(26,\cdot)\) \(\chi_{4447}(32,\cdot)\) \(\chi_{4447}(35,\cdot)\) \(\chi_{4447}(36,\cdot)\) \(\chi_{4447}(37,\cdot)\) \(\chi_{4447}(42,\cdot)\) \(\chi_{4447}(49,\cdot)\) \(\chi_{4447}(50,\cdot)\) \(\chi_{4447}(51,\cdot)\) \(\chi_{4447}(52,\cdot)\) \(\chi_{4447}(55,\cdot)\) \(\chi_{4447}(57,\cdot)\) \(\chi_{4447}(66,\cdot)\) \(\chi_{4447}(69,\cdot)\) \(\chi_{4447}(74,\cdot)\) \(\chi_{4447}(77,\cdot)\) \(\chi_{4447}(81,\cdot)\) \(\chi_{4447}(85,\cdot)\) \(\chi_{4447}(87,\cdot)\) \(\chi_{4447}(93,\cdot)\) \(\chi_{4447}(94,\cdot)\) \(\chi_{4447}(98,\cdot)\) \(\chi_{4447}(100,\cdot)\) \(\chi_{4447}(107,\cdot)\) \(\chi_{4447}(109,\cdot)\) ...

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: $\Q(\zeta_{2223})$
Fixed field: Number field defined by a degree 2223 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1490}{2223}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4447 }(1042, a) \) \(1\)\(1\)\(e\left(\frac{989}{2223}\right)\)\(e\left(\frac{1490}{2223}\right)\)\(e\left(\frac{1978}{2223}\right)\)\(e\left(\frac{181}{247}\right)\)\(e\left(\frac{256}{2223}\right)\)\(e\left(\frac{1654}{2223}\right)\)\(e\left(\frac{248}{741}\right)\)\(e\left(\frac{757}{2223}\right)\)\(e\left(\frac{395}{2223}\right)\)\(e\left(\frac{2047}{2223}\right)\)
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)
 
\( \chi_{ 4447 }(1042,a) \;\) at \(\;a = \) e.g. 2