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Properties

Label 53176.861
Modulus $53176$
Conductor $53176$
Order $2992$
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(53176, base_ring=CyclotomicField(2992)) M = H._module chi = DirichletCharacter(H, M([0,1496,605,408]))
 
Copy content gp:[g,chi] = znchar(Mod(861, 53176))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("53176.861");
 

Basic properties

Modulus: \(53176\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(53176\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2992\)
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 53176.gb

\(\chi_{53176}(5,\cdot)\) \(\chi_{53176}(37,\cdot)\) \(\chi_{53176}(61,\cdot)\) \(\chi_{53176}(109,\cdot)\) \(\chi_{53176}(125,\cdot)\) \(\chi_{53176}(181,\cdot)\) \(\chi_{53176}(245,\cdot)\) \(\chi_{53176}(309,\cdot)\) \(\chi_{53176}(333,\cdot)\) \(\chi_{53176}(405,\cdot)\) \(\chi_{53176}(517,\cdot)\) \(\chi_{53176}(549,\cdot)\) \(\chi_{53176}(573,\cdot)\) \(\chi_{53176}(589,\cdot)\) \(\chi_{53176}(605,\cdot)\) \(\chi_{53176}(677,\cdot)\) \(\chi_{53176}(741,\cdot)\) \(\chi_{53176}(789,\cdot)\) \(\chi_{53176}(845,\cdot)\) \(\chi_{53176}(861,\cdot)\) \(\chi_{53176}(925,\cdot)\) \(\chi_{53176}(941,\cdot)\) \(\chi_{53176}(957,\cdot)\) \(\chi_{53176}(981,\cdot)\) \(\chi_{53176}(1077,\cdot)\) \(\chi_{53176}(1125,\cdot)\) \(\chi_{53176}(1213,\cdot)\) \(\chi_{53176}(1229,\cdot)\) \(\chi_{53176}(1253,\cdot)\) \(\chi_{53176}(1261,\cdot)\) ...

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: $\Q(\zeta_{2992})$
Fixed field: Number field defined by a degree 2992 polynomial (not computed)

Values on generators

\((13295,26589,4049,18497)\) → \((1,-1,e\left(\frac{55}{272}\right),e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(25\)
\( \chi_{ 53176 }(861, a) \) \(1\)\(1\)\(e\left(\frac{2645}{2992}\right)\)\(e\left(\frac{2817}{2992}\right)\)\(e\left(\frac{2791}{2992}\right)\)\(e\left(\frac{1149}{1496}\right)\)\(e\left(\frac{1131}{2992}\right)\)\(e\left(\frac{31}{748}\right)\)\(e\left(\frac{1235}{1496}\right)\)\(e\left(\frac{563}{1496}\right)\)\(e\left(\frac{611}{748}\right)\)\(e\left(\frac{1321}{1496}\right)\)
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)
 
\( \chi_{ 53176 }(861,a) \;\) at \(\;a = \) e.g. 2