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Properties

Label 12000.2759
Modulus $12000$
Conductor $6000$
Order $100$
Real no
Primitive no
Minimal no
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(12000, base_ring=CyclotomicField(100)) M = H._module chi = DirichletCharacter(H, M([50,25,50,14]))
 
Copy content gp:[g,chi] = znchar(Mod(2759, 12000))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12000.2759");
 

Basic properties

Modulus: \(12000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6000\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(100\)
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 \(\chi_{6000}(1259,\cdot)\)
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: no
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 12000.fu

\(\chi_{12000}(119,\cdot)\) \(\chi_{12000}(359,\cdot)\) \(\chi_{12000}(839,\cdot)\) \(\chi_{12000}(1079,\cdot)\) \(\chi_{12000}(1319,\cdot)\) \(\chi_{12000}(1559,\cdot)\) \(\chi_{12000}(2039,\cdot)\) \(\chi_{12000}(2279,\cdot)\) \(\chi_{12000}(2519,\cdot)\) \(\chi_{12000}(2759,\cdot)\) \(\chi_{12000}(3239,\cdot)\) \(\chi_{12000}(3479,\cdot)\) \(\chi_{12000}(3719,\cdot)\) \(\chi_{12000}(3959,\cdot)\) \(\chi_{12000}(4439,\cdot)\) \(\chi_{12000}(4679,\cdot)\) \(\chi_{12000}(4919,\cdot)\) \(\chi_{12000}(5159,\cdot)\) \(\chi_{12000}(5639,\cdot)\) \(\chi_{12000}(5879,\cdot)\) \(\chi_{12000}(6119,\cdot)\) \(\chi_{12000}(6359,\cdot)\) \(\chi_{12000}(6839,\cdot)\) \(\chi_{12000}(7079,\cdot)\) \(\chi_{12000}(7319,\cdot)\) \(\chi_{12000}(7559,\cdot)\) \(\chi_{12000}(8039,\cdot)\) \(\chi_{12000}(8279,\cdot)\) \(\chi_{12000}(8519,\cdot)\) \(\chi_{12000}(8759,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((6751,10501,4001,5377)\) → \((-1,i,-1,e\left(\frac{7}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 12000 }(2759, a) \) \(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{89}{100}\right)\)\(e\left(\frac{21}{100}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{77}{100}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{93}{100}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{31}{100}\right)\)\(e\left(\frac{4}{25}\right)\)
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)
 
\( \chi_{ 12000 }(2759,a) \;\) at \(\;a = \) e.g. 2