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Properties

Label 15824.5941
Modulus $15824$
Conductor $15824$
Order $132$
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(15824, base_ring=CyclotomicField(132)) M = H._module chi = DirichletCharacter(H, M([0,33,114,110]))
 
Copy content gp:[g,chi] = znchar(Mod(5941, 15824))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15824.5941");
 

Basic properties

Modulus: \(15824\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(15824\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(132\)
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 15824.fh

\(\chi_{15824}(37,\cdot)\) \(\chi_{15824}(1069,\cdot)\) \(\chi_{15824}(1125,\cdot)\) \(\chi_{15824}(1413,\cdot)\) \(\chi_{15824}(1469,\cdot)\) \(\chi_{15824}(1813,\cdot)\) \(\chi_{15824}(2445,\cdot)\) \(\chi_{15824}(2501,\cdot)\) \(\chi_{15824}(3133,\cdot)\) \(\chi_{15824}(3189,\cdot)\) \(\chi_{15824}(3533,\cdot)\) \(\chi_{15824}(4565,\cdot)\) \(\chi_{15824}(4909,\cdot)\) \(\chi_{15824}(5541,\cdot)\) \(\chi_{15824}(5885,\cdot)\) \(\chi_{15824}(5941,\cdot)\) \(\chi_{15824}(6229,\cdot)\) \(\chi_{15824}(6629,\cdot)\) \(\chi_{15824}(6917,\cdot)\) \(\chi_{15824}(7605,\cdot)\) \(\chi_{15824}(7949,\cdot)\) \(\chi_{15824}(8981,\cdot)\) \(\chi_{15824}(9037,\cdot)\) \(\chi_{15824}(9325,\cdot)\) \(\chi_{15824}(9381,\cdot)\) \(\chi_{15824}(9725,\cdot)\) \(\chi_{15824}(10357,\cdot)\) \(\chi_{15824}(10413,\cdot)\) \(\chi_{15824}(11045,\cdot)\) \(\chi_{15824}(11101,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((5935,3957,14449,6625)\) → \((1,i,e\left(\frac{19}{22}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 15824 }(5941, a) \) \(1\)\(1\)\(e\left(\frac{53}{132}\right)\)\(e\left(\frac{125}{132}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{67}{132}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{71}{132}\right)\)\(e\left(\frac{21}{44}\right)\)
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)
 
\( \chi_{ 15824 }(5941,a) \;\) at \(\;a = \) e.g. 2