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Properties

Label 56629.94
Modulus $56629$
Conductor $56629$
Order $28314$
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(56629, base_ring=CyclotomicField(28314)) M = H._module chi = DirichletCharacter(H, M([6191]))
 
Copy content gp:[g,chi] = znchar(Mod(94, 56629))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("56629.94");
 

Basic properties

Modulus: \(56629\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(56629\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(28314\)
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 56629.ca

\(\chi_{56629}(4,\cdot)\) \(\chi_{56629}(5,\cdot)\) \(\chi_{56629}(11,\cdot)\) \(\chi_{56629}(36,\cdot)\) \(\chi_{56629}(39,\cdot)\) \(\chi_{56629}(42,\cdot)\) \(\chi_{56629}(45,\cdot)\) \(\chi_{56629}(48,\cdot)\) \(\chi_{56629}(49,\cdot)\) \(\chi_{56629}(51,\cdot)\) \(\chi_{56629}(52,\cdot)\) \(\chi_{56629}(56,\cdot)\) \(\chi_{56629}(58,\cdot)\) \(\chi_{56629}(60,\cdot)\) \(\chi_{56629}(68,\cdot)\) \(\chi_{56629}(73,\cdot)\) \(\chi_{56629}(75,\cdot)\) \(\chi_{56629}(79,\cdot)\) \(\chi_{56629}(85,\cdot)\) \(\chi_{56629}(94,\cdot)\) \(\chi_{56629}(103,\cdot)\) \(\chi_{56629}(122,\cdot)\) \(\chi_{56629}(123,\cdot)\) \(\chi_{56629}(129,\cdot)\) \(\chi_{56629}(142,\cdot)\) \(\chi_{56629}(157,\cdot)\) \(\chi_{56629}(171,\cdot)\) \(\chi_{56629}(176,\cdot)\) \(\chi_{56629}(186,\cdot)\) \(\chi_{56629}(201,\cdot)\) ...

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: $\Q(\zeta_{14157})$
Fixed field: Number field defined by a degree 28314 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{6191}{28314}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 56629 }(94, a) \) \(1\)\(1\)\(e\left(\frac{6191}{28314}\right)\)\(e\left(\frac{4234}{4719}\right)\)\(e\left(\frac{6191}{14157}\right)\)\(e\left(\frac{356}{14157}\right)\)\(e\left(\frac{3281}{28314}\right)\)\(e\left(\frac{5549}{28314}\right)\)\(e\left(\frac{6191}{9438}\right)\)\(e\left(\frac{3749}{4719}\right)\)\(e\left(\frac{59}{242}\right)\)\(e\left(\frac{10687}{14157}\right)\)
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)
 
\( \chi_{ 56629 }(94,a) \;\) at \(\;a = \) e.g. 2