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Properties

Label 11012.259
Modulus $11012$
Conductor $11012$
Order $2752$
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(11012, base_ring=CyclotomicField(2752)) M = H._module chi = DirichletCharacter(H, M([1376,1733]))
 
Copy content gp:[g,chi] = znchar(Mod(259, 11012))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11012.259");
 

Basic properties

Modulus: \(11012\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11012\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2752\)
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 11012.ba

\(\chi_{11012}(3,\cdot)\) \(\chi_{11012}(27,\cdot)\) \(\chi_{11012}(35,\cdot)\) \(\chi_{11012}(39,\cdot)\) \(\chi_{11012}(51,\cdot)\) \(\chi_{11012}(55,\cdot)\) \(\chi_{11012}(59,\cdot)\) \(\chi_{11012}(71,\cdot)\) \(\chi_{11012}(75,\cdot)\) \(\chi_{11012}(83,\cdot)\) \(\chi_{11012}(95,\cdot)\) \(\chi_{11012}(103,\cdot)\) \(\chi_{11012}(107,\cdot)\) \(\chi_{11012}(115,\cdot)\) \(\chi_{11012}(127,\cdot)\) \(\chi_{11012}(131,\cdot)\) \(\chi_{11012}(147,\cdot)\) \(\chi_{11012}(151,\cdot)\) \(\chi_{11012}(155,\cdot)\) \(\chi_{11012}(199,\cdot)\) \(\chi_{11012}(203,\cdot)\) \(\chi_{11012}(211,\cdot)\) \(\chi_{11012}(215,\cdot)\) \(\chi_{11012}(235,\cdot)\) \(\chi_{11012}(243,\cdot)\) \(\chi_{11012}(259,\cdot)\) \(\chi_{11012}(263,\cdot)\) \(\chi_{11012}(271,\cdot)\) \(\chi_{11012}(283,\cdot)\) \(\chi_{11012}(287,\cdot)\) ...

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: $\Q(\zeta_{2752})$
Fixed field: Number field defined by a degree 2752 polynomial (not computed)

Values on generators

\((5507,5509)\) → \((-1,e\left(\frac{1733}{2752}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 11012 }(259, a) \) \(1\)\(1\)\(e\left(\frac{357}{2752}\right)\)\(e\left(\frac{799}{2752}\right)\)\(e\left(\frac{525}{688}\right)\)\(e\left(\frac{357}{1376}\right)\)\(e\left(\frac{599}{1376}\right)\)\(e\left(\frac{63}{172}\right)\)\(e\left(\frac{289}{688}\right)\)\(e\left(\frac{217}{1376}\right)\)\(e\left(\frac{26}{43}\right)\)\(e\left(\frac{2457}{2752}\right)\)
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)
 
\( \chi_{ 11012 }(259,a) \;\) at \(\;a = \) e.g. 2