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Properties

Label 2381.2
Modulus $2381$
Conductor $2381$
Order $476$
Real no
Primitive yes
Minimal yes
Parity odd

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(2381, base_ring=CyclotomicField(476)) M = H._module chi = DirichletCharacter(H, M([181]))
 
Copy content gp:[g,chi] = znchar(Mod(2, 2381))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2381.2");
 

Basic properties

Modulus: \(2381\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2381\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(476\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 2381.u

\(\chi_{2381}(2,\cdot)\) \(\chi_{2381}(8,\cdot)\) \(\chi_{2381}(10,\cdot)\) \(\chi_{2381}(14,\cdot)\) \(\chi_{2381}(32,\cdot)\) \(\chi_{2381}(34,\cdot)\) \(\chi_{2381}(40,\cdot)\) \(\chi_{2381}(50,\cdot)\) \(\chi_{2381}(56,\cdot)\) \(\chi_{2381}(70,\cdot)\) \(\chi_{2381}(98,\cdot)\) \(\chi_{2381}(136,\cdot)\) \(\chi_{2381}(143,\cdot)\) \(\chi_{2381}(164,\cdot)\) \(\chi_{2381}(170,\cdot)\) \(\chi_{2381}(179,\cdot)\) \(\chi_{2381}(205,\cdot)\) \(\chi_{2381}(223,\cdot)\) \(\chi_{2381}(224,\cdot)\) \(\chi_{2381}(238,\cdot)\) \(\chi_{2381}(243,\cdot)\) \(\chi_{2381}(267,\cdot)\) \(\chi_{2381}(276,\cdot)\) \(\chi_{2381}(280,\cdot)\) \(\chi_{2381}(282,\cdot)\) \(\chi_{2381}(287,\cdot)\) \(\chi_{2381}(297,\cdot)\) \(\chi_{2381}(333,\cdot)\) \(\chi_{2381}(339,\cdot)\) \(\chi_{2381}(345,\cdot)\) ...

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: $\Q(\zeta_{476})$
Fixed field: Number field defined by a degree 476 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{181}{476}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2381 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{61}{476}\right)\)\(e\left(\frac{181}{476}\right)\)\(e\left(\frac{61}{238}\right)\)\(e\left(\frac{47}{238}\right)\)\(e\left(\frac{121}{238}\right)\)\(e\left(\frac{27}{238}\right)\)\(e\left(\frac{183}{476}\right)\)\(e\left(\frac{181}{238}\right)\)\(e\left(\frac{155}{476}\right)\)\(e\left(\frac{41}{238}\right)\)
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)
 
\( \chi_{ 2381 }(2,a) \;\) at \(\;a = \) e.g. 2