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Properties

Label 11571.1517
Modulus $11571$
Conductor $11571$
Order $126$
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(11571, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([63,105,28,45]))
 
Copy content gp:[g,chi] = znchar(Mod(1517, 11571))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11571.1517");
 

Basic properties

Modulus: \(11571\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11571\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(126\)
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 11571.nb

\(\chi_{11571}(5,\cdot)\) \(\chi_{11571}(80,\cdot)\) \(\chi_{11571}(332,\cdot)\) \(\chi_{11571}(731,\cdot)\) \(\chi_{11571}(1202,\cdot)\) \(\chi_{11571}(1298,\cdot)\) \(\chi_{11571}(1517,\cdot)\) \(\chi_{11571}(1601,\cdot)\) \(\chi_{11571}(2126,\cdot)\) \(\chi_{11571}(2474,\cdot)\) \(\chi_{11571}(3125,\cdot)\) \(\chi_{11571}(3692,\cdot)\) \(\chi_{11571}(3995,\cdot)\) \(\chi_{11571}(5108,\cdot)\) \(\chi_{11571}(5519,\cdot)\) \(\chi_{11571}(5717,\cdot)\) \(\chi_{11571}(6065,\cdot)\) \(\chi_{11571}(6389,\cdot)\) \(\chi_{11571}(6704,\cdot)\) \(\chi_{11571}(7283,\cdot)\) \(\chi_{11571}(7313,\cdot)\) \(\chi_{11571}(7661,\cdot)\) \(\chi_{11571}(7901,\cdot)\) \(\chi_{11571}(8300,\cdot)\) \(\chi_{11571}(8510,\cdot)\) \(\chi_{11571}(8858,\cdot)\) \(\chi_{11571}(8879,\cdot)\) \(\chi_{11571}(8909,\cdot)\) \(\chi_{11571}(9110,\cdot)\) \(\chi_{11571}(9257,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((7715,3307,610,1597)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{2}{9}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(20\)
\( \chi_{ 11571 }(1517, a) \) \(1\)\(1\)\(e\left(\frac{47}{63}\right)\)\(e\left(\frac{31}{63}\right)\)\(e\left(\frac{5}{63}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{52}{63}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{126}\right)\)\(e\left(\frac{62}{63}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{7}\right)\)
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)
 
\( \chi_{ 11571 }(1517,a) \;\) at \(\;a = \) e.g. 2