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Properties

Label 5077.1005
Modulus $5077$
Conductor $5077$
Order $2538$
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(5077, base_ring=CyclotomicField(2538)) M = H._module chi = DirichletCharacter(H, M([2203]))
 
Copy content gp:[g,chi] = znchar(Mod(1005, 5077))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5077.1005");
 

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: \(2538\)
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 5077.w

\(\chi_{5077}(4,\cdot)\) \(\chi_{5077}(10,\cdot)\) \(\chi_{5077}(19,\cdot)\) \(\chi_{5077}(34,\cdot)\) \(\chi_{5077}(36,\cdot)\) \(\chi_{5077}(47,\cdot)\) \(\chi_{5077}(48,\cdot)\) \(\chi_{5077}(55,\cdot)\) \(\chi_{5077}(58,\cdot)\) \(\chi_{5077}(61,\cdot)\) \(\chi_{5077}(70,\cdot)\) \(\chi_{5077}(71,\cdot)\) \(\chi_{5077}(78,\cdot)\) \(\chi_{5077}(79,\cdot)\) \(\chi_{5077}(85,\cdot)\) \(\chi_{5077}(90,\cdot)\) \(\chi_{5077}(103,\cdot)\) \(\chi_{5077}(121,\cdot)\) \(\chi_{5077}(137,\cdot)\) \(\chi_{5077}(138,\cdot)\) \(\chi_{5077}(146,\cdot)\) \(\chi_{5077}(147,\cdot)\) \(\chi_{5077}(154,\cdot)\) \(\chi_{5077}(160,\cdot)\) \(\chi_{5077}(166,\cdot)\) \(\chi_{5077}(171,\cdot)\) \(\chi_{5077}(173,\cdot)\) \(\chi_{5077}(175,\cdot)\) \(\chi_{5077}(177,\cdot)\) \(\chi_{5077}(186,\cdot)\) ...

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: $\Q(\zeta_{1269})$
Fixed field: Number field defined by a degree 2538 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{2203}{2538}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5077 }(1005, a) \) \(1\)\(1\)\(e\left(\frac{2203}{2538}\right)\)\(e\left(\frac{103}{423}\right)\)\(e\left(\frac{934}{1269}\right)\)\(e\left(\frac{75}{94}\right)\)\(e\left(\frac{283}{2538}\right)\)\(e\left(\frac{1148}{1269}\right)\)\(e\left(\frac{511}{846}\right)\)\(e\left(\frac{206}{423}\right)\)\(e\left(\frac{845}{1269}\right)\)\(e\left(\frac{2261}{2538}\right)\)
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)
 
\( \chi_{ 5077 }(1005,a) \;\) at \(\;a = \) e.g. 2