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Properties

Label 58492.69
Modulus $58492$
Conductor $14623$
Order $2088$
Real no
Primitive no
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(58492, base_ring=CyclotomicField(2088)) M = H._module chi = DirichletCharacter(H, M([0,1044,2063]))
 
Copy content gp:[g,chi] = znchar(Mod(69, 58492))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("58492.69");
 

Basic properties

Modulus: \(58492\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(14623\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2088\)
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: no, induced from \(\chi_{14623}(69,\cdot)\)
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 58492.jp

\(\chi_{58492}(69,\cdot)\) \(\chi_{58492}(97,\cdot)\) \(\chi_{58492}(265,\cdot)\) \(\chi_{58492}(433,\cdot)\) \(\chi_{58492}(601,\cdot)\) \(\chi_{58492}(629,\cdot)\) \(\chi_{58492}(741,\cdot)\) \(\chi_{58492}(769,\cdot)\) \(\chi_{58492}(797,\cdot)\) \(\chi_{58492}(825,\cdot)\) \(\chi_{58492}(909,\cdot)\) \(\chi_{58492}(965,\cdot)\) \(\chi_{58492}(1133,\cdot)\) \(\chi_{58492}(1217,\cdot)\) \(\chi_{58492}(1245,\cdot)\) \(\chi_{58492}(1329,\cdot)\) \(\chi_{58492}(1385,\cdot)\) \(\chi_{58492}(1469,\cdot)\)

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: $\Q(\zeta_{2088})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 2088 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((29247,50137,54321)\) → \((1,-1,e\left(\frac{2063}{2088}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 58492 }(69, a) \) \(1\)\(1\)\(e\left(\frac{215}{1044}\right)\)\(e\left(\frac{245}{1044}\right)\)\(e\left(\frac{215}{522}\right)\)\(e\left(\frac{197}{2088}\right)\)\(e\left(\frac{887}{1044}\right)\)\(e\left(\frac{115}{261}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{491}{696}\right)\)\(e\left(\frac{413}{696}\right)\)\(e\left(\frac{245}{522}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 58492 }(69,a) \;\) at \(\;a = \) e.g. 2