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Properties

Label 1333.293
Modulus $1333$
Conductor $1333$
Order $105$
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(1333, base_ring=CyclotomicField(210)) M = H._module chi = DirichletCharacter(H, M([154,90]))
 
Copy content gp:[g,chi] = znchar(Mod(293, 1333))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1333.293");
 

Basic properties

Modulus: \(1333\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1333\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(105\)
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 1333.cn

\(\chi_{1333}(41,\cdot)\) \(\chi_{1333}(59,\cdot)\) \(\chi_{1333}(90,\cdot)\) \(\chi_{1333}(102,\cdot)\) \(\chi_{1333}(107,\cdot)\) \(\chi_{1333}(121,\cdot)\) \(\chi_{1333}(133,\cdot)\) \(\chi_{1333}(164,\cdot)\) \(\chi_{1333}(183,\cdot)\) \(\chi_{1333}(193,\cdot)\) \(\chi_{1333}(226,\cdot)\) \(\chi_{1333}(231,\cdot)\) \(\chi_{1333}(236,\cdot)\) \(\chi_{1333}(262,\cdot)\) \(\chi_{1333}(293,\cdot)\) \(\chi_{1333}(299,\cdot)\) \(\chi_{1333}(317,\cdot)\) \(\chi_{1333}(348,\cdot)\) \(\chi_{1333}(355,\cdot)\) \(\chi_{1333}(360,\cdot)\) \(\chi_{1333}(379,\cdot)\) \(\chi_{1333}(391,\cdot)\) \(\chi_{1333}(422,\cdot)\) \(\chi_{1333}(441,\cdot)\) \(\chi_{1333}(484,\cdot)\) \(\chi_{1333}(514,\cdot)\) \(\chi_{1333}(537,\cdot)\) \(\chi_{1333}(661,\cdot)\) \(\chi_{1333}(692,\cdot)\) \(\chi_{1333}(723,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((1119,218)\) → \((e\left(\frac{11}{15}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1333 }(293, a) \) \(1\)\(1\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{17}{105}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{34}{105}\right)\)\(e\left(\frac{58}{105}\right)\)\(e\left(\frac{76}{105}\right)\)
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)
 
\( \chi_{ 1333 }(293,a) \;\) at \(\;a = \) e.g. 2