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Properties

Label 11191.131
Modulus $11191$
Conductor $11191$
Order $855$
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(11191, base_ring=CyclotomicField(1710)) M = H._module chi = DirichletCharacter(H, M([1040,1596]))
 
Copy content gp:[g,chi] = znchar(Mod(131, 11191))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11191.131");
 

Basic properties

Modulus: \(11191\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11191\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(855\)
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 11191.em

\(\chi_{11191}(9,\cdot)\) \(\chi_{11191}(80,\cdot)\) \(\chi_{11191}(81,\cdot)\) \(\chi_{11191}(82,\cdot)\) \(\chi_{11191}(100,\cdot)\) \(\chi_{11191}(112,\cdot)\) \(\chi_{11191}(131,\cdot)\) \(\chi_{11191}(138,\cdot)\) \(\chi_{11191}(142,\cdot)\) \(\chi_{11191}(169,\cdot)\) \(\chi_{11191}(175,\cdot)\) \(\chi_{11191}(196,\cdot)\) \(\chi_{11191}(214,\cdot)\) \(\chi_{11191}(237,\cdot)\) \(\chi_{11191}(289,\cdot)\) \(\chi_{11191}(351,\cdot)\) \(\chi_{11191}(443,\cdot)\) \(\chi_{11191}(453,\cdot)\) \(\chi_{11191}(472,\cdot)\) \(\chi_{11191}(510,\cdot)\) \(\chi_{11191}(536,\cdot)\) \(\chi_{11191}(576,\cdot)\) \(\chi_{11191}(586,\cdot)\) \(\chi_{11191}(598,\cdot)\) \(\chi_{11191}(669,\cdot)\) \(\chi_{11191}(670,\cdot)\) \(\chi_{11191}(671,\cdot)\) \(\chi_{11191}(689,\cdot)\) \(\chi_{11191}(701,\cdot)\) \(\chi_{11191}(720,\cdot)\) ...

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: $\Q(\zeta_{855})$
Fixed field: Number field defined by a degree 855 polynomial (not computed)

Values on generators

\((6139,10109)\) → \((e\left(\frac{104}{171}\right),e\left(\frac{14}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 11191 }(131, a) \) \(1\)\(1\)\(e\left(\frac{7}{855}\right)\)\(e\left(\frac{403}{855}\right)\)\(e\left(\frac{14}{855}\right)\)\(e\left(\frac{131}{171}\right)\)\(e\left(\frac{82}{171}\right)\)\(e\left(\frac{103}{285}\right)\)\(e\left(\frac{7}{285}\right)\)\(e\left(\frac{806}{855}\right)\)\(e\left(\frac{662}{855}\right)\)\(e\left(\frac{143}{285}\right)\)
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)
 
\( \chi_{ 11191 }(131,a) \;\) at \(\;a = \) e.g. 2