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Properties

Label 12441.1367
Modulus $12441$
Conductor $12441$
Order $420$
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(12441, base_ring=CyclotomicField(420)) M = H._module chi = DirichletCharacter(H, M([210,336,35,30]))
 
Copy content gp:[g,chi] = znchar(Mod(1367, 12441))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12441.1367");
 

Basic properties

Modulus: \(12441\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12441\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(420\)
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 12441.ly

\(\chi_{12441}(71,\cdot)\) \(\chi_{12441}(80,\cdot)\) \(\chi_{12441}(158,\cdot)\) \(\chi_{12441}(236,\cdot)\) \(\chi_{12441}(245,\cdot)\) \(\chi_{12441}(323,\cdot)\) \(\chi_{12441}(383,\cdot)\) \(\chi_{12441}(410,\cdot)\) \(\chi_{12441}(995,\cdot)\) \(\chi_{12441}(1202,\cdot)\) \(\chi_{12441}(1280,\cdot)\) \(\chi_{12441}(1367,\cdot)\) \(\chi_{12441}(1571,\cdot)\) \(\chi_{12441}(1688,\cdot)\) \(\chi_{12441}(1775,\cdot)\) \(\chi_{12441}(1952,\cdot)\) \(\chi_{12441}(2039,\cdot)\) \(\chi_{12441}(2126,\cdot)\) \(\chi_{12441}(2429,\cdot)\) \(\chi_{12441}(2528,\cdot)\) \(\chi_{12441}(2594,\cdot)\) \(\chi_{12441}(2645,\cdot)\) \(\chi_{12441}(2732,\cdot)\) \(\chi_{12441}(2819,\cdot)\) \(\chi_{12441}(2996,\cdot)\) \(\chi_{12441}(3083,\cdot)\) \(\chi_{12441}(3386,\cdot)\) \(\chi_{12441}(3551,\cdot)\) \(\chi_{12441}(3716,\cdot)\) \(\chi_{12441}(3776,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((4148,5656,4786,10297)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{1}{12}\right),e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 12441 }(1367, a) \) \(1\)\(1\)\(e\left(\frac{191}{420}\right)\)\(e\left(\frac{191}{210}\right)\)\(e\left(\frac{3}{140}\right)\)\(e\left(\frac{157}{420}\right)\)\(e\left(\frac{51}{140}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{86}{105}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{193}{420}\right)\)
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)
 
\( \chi_{ 12441 }(1367,a) \;\) at \(\;a = \) e.g. 2