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Properties

Label 22925.13082
Modulus $22925$
Conductor $4585$
Order $52$
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(22925, base_ring=CyclotomicField(52)) M = H._module chi = DirichletCharacter(H, M([13,26,32]))
 
Copy content gp:[g,chi] = znchar(Mod(13082, 22925))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("22925.13082");
 

Basic properties

Modulus: \(22925\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4585\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(52\)
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 \(\chi_{4585}(3912,\cdot)\)
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 22925.ga

\(\chi_{22925}(307,\cdot)\) \(\chi_{22925}(3093,\cdot)\) \(\chi_{22925}(3982,\cdot)\) \(\chi_{22925}(4493,\cdot)\) \(\chi_{22925}(5193,\cdot)\) \(\chi_{22925}(5732,\cdot)\) \(\chi_{22925}(7643,\cdot)\) \(\chi_{22925}(8182,\cdot)\) \(\chi_{22925}(9232,\cdot)\) \(\chi_{22925}(9757,\cdot)\) \(\chi_{22925}(9932,\cdot)\) \(\chi_{22925}(11318,\cdot)\) \(\chi_{22925}(12557,\cdot)\) \(\chi_{22925}(13068,\cdot)\) \(\chi_{22925}(13082,\cdot)\) \(\chi_{22925}(15518,\cdot)\) \(\chi_{22925}(16568,\cdot)\) \(\chi_{22925}(17093,\cdot)\) \(\chi_{22925}(17268,\cdot)\) \(\chi_{22925}(18682,\cdot)\) \(\chi_{22925}(19893,\cdot)\) \(\chi_{22925}(20082,\cdot)\) \(\chi_{22925}(20418,\cdot)\) \(\chi_{22925}(20782,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((2752,16376,526)\) → \((i,-1,e\left(\frac{8}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 22925 }(13082, a) \) \(1\)\(1\)\(e\left(\frac{45}{52}\right)\)\(e\left(\frac{29}{52}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{15}{52}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{6}{13}\right)\)
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)
 
\( \chi_{ 22925 }(13082,a) \;\) at \(\;a = \) e.g. 2