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Properties

Label 4027.248
Modulus $4027$
Conductor $4027$
Order $183$
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(4027, base_ring=CyclotomicField(366)) M = H._module chi = DirichletCharacter(H, M([58]))
 
Copy content gp:[g,chi] = znchar(Mod(248, 4027))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4027.248");
 

Basic properties

Modulus: \(4027\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4027\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(183\)
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 4027.k

\(\chi_{4027}(41,\cdot)\) \(\chi_{4027}(68,\cdot)\) \(\chi_{4027}(102,\cdot)\) \(\chi_{4027}(140,\cdot)\) \(\chi_{4027}(155,\cdot)\) \(\chi_{4027}(218,\cdot)\) \(\chi_{4027}(248,\cdot)\) \(\chi_{4027}(308,\cdot)\) \(\chi_{4027}(315,\cdot)\) \(\chi_{4027}(334,\cdot)\) \(\chi_{4027}(336,\cdot)\) \(\chi_{4027}(341,\cdot)\) \(\chi_{4027}(397,\cdot)\) \(\chi_{4027}(398,\cdot)\) \(\chi_{4027}(406,\cdot)\) \(\chi_{4027}(489,\cdot)\) \(\chi_{4027}(501,\cdot)\) \(\chi_{4027}(504,\cdot)\) \(\chi_{4027}(508,\cdot)\) \(\chi_{4027}(529,\cdot)\) \(\chi_{4027}(533,\cdot)\) \(\chi_{4027}(547,\cdot)\) \(\chi_{4027}(556,\cdot)\) \(\chi_{4027}(558,\cdot)\) \(\chi_{4027}(597,\cdot)\) \(\chi_{4027}(599,\cdot)\) \(\chi_{4027}(609,\cdot)\) \(\chi_{4027}(693,\cdot)\) \(\chi_{4027}(807,\cdot)\) \(\chi_{4027}(837,\cdot)\) ...

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: $\Q(\zeta_{183})$
Fixed field: Number field defined by a degree 183 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{29}{183}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4027 }(248, a) \) \(1\)\(1\)\(e\left(\frac{26}{61}\right)\)\(e\left(\frac{29}{183}\right)\)\(e\left(\frac{52}{61}\right)\)\(e\left(\frac{82}{183}\right)\)\(e\left(\frac{107}{183}\right)\)\(e\left(\frac{35}{61}\right)\)\(e\left(\frac{17}{61}\right)\)\(e\left(\frac{58}{183}\right)\)\(e\left(\frac{160}{183}\right)\)\(e\left(\frac{13}{183}\right)\)
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)
 
\( \chi_{ 4027 }(248,a) \;\) at \(\;a = \) e.g. 2