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Properties

Label 5077.492
Modulus $5077$
Conductor $5077$
Order $564$
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(5077, base_ring=CyclotomicField(564)) M = H._module chi = DirichletCharacter(H, M([307]))
 
Copy content gp:[g,chi] = znchar(Mod(492, 5077))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5077.492");
 

Basic properties

Modulus: \(5077\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5077\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(564\)
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 5077.s

\(\chi_{5077}(24,\cdot)\) \(\chi_{5077}(39,\cdot)\) \(\chi_{5077}(98,\cdot)\) \(\chi_{5077}(110,\cdot)\) \(\chi_{5077}(135,\cdot)\) \(\chi_{5077}(149,\cdot)\) \(\chi_{5077}(207,\cdot)\) \(\chi_{5077}(229,\cdot)\) \(\chi_{5077}(231,\cdot)\) \(\chi_{5077}(251,\cdot)\) \(\chi_{5077}(261,\cdot)\) \(\chi_{5077}(326,\cdot)\) \(\chi_{5077}(372,\cdot)\) \(\chi_{5077}(397,\cdot)\) \(\chi_{5077}(424,\cdot)\) \(\chi_{5077}(445,\cdot)\) \(\chi_{5077}(446,\cdot)\) \(\chi_{5077}(466,\cdot)\) \(\chi_{5077}(485,\cdot)\) \(\chi_{5077}(492,\cdot)\) \(\chi_{5077}(512,\cdot)\) \(\chi_{5077}(545,\cdot)\) \(\chi_{5077}(600,\cdot)\) \(\chi_{5077}(648,\cdot)\) \(\chi_{5077}(661,\cdot)\) \(\chi_{5077}(665,\cdot)\) \(\chi_{5077}(670,\cdot)\) \(\chi_{5077}(689,\cdot)\) \(\chi_{5077}(698,\cdot)\) \(\chi_{5077}(718,\cdot)\) ...

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: $\Q(\zeta_{564})$
Fixed field: Number field defined by a degree 564 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{307}{564}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5077 }(492, a) \) \(-1\)\(1\)\(e\left(\frac{307}{564}\right)\)\(e\left(\frac{33}{94}\right)\)\(e\left(\frac{25}{282}\right)\)\(e\left(\frac{125}{188}\right)\)\(e\left(\frac{505}{564}\right)\)\(e\left(\frac{115}{141}\right)\)\(e\left(\frac{119}{188}\right)\)\(e\left(\frac{33}{47}\right)\)\(e\left(\frac{59}{282}\right)\)\(e\left(\frac{551}{564}\right)\)
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)
 
\( \chi_{ 5077 }(492,a) \;\) at \(\;a = \) e.g. 2