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Properties

Label 1132.139
Modulus $1132$
Conductor $1132$
Order $282$
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(1132, base_ring=CyclotomicField(282)) M = H._module chi = DirichletCharacter(H, M([141,215]))
 
Copy content gp:[g,chi] = znchar(Mod(139, 1132))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1132.139");
 

Basic properties

Modulus: \(1132\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1132\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(282\)
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 1132.p

\(\chi_{1132}(3,\cdot)\) \(\chi_{1132}(31,\cdot)\) \(\chi_{1132}(35,\cdot)\) \(\chi_{1132}(47,\cdot)\) \(\chi_{1132}(55,\cdot)\) \(\chi_{1132}(75,\cdot)\) \(\chi_{1132}(87,\cdot)\) \(\chi_{1132}(107,\cdot)\) \(\chi_{1132}(119,\cdot)\) \(\chi_{1132}(123,\cdot)\) \(\chi_{1132}(139,\cdot)\) \(\chi_{1132}(147,\cdot)\) \(\chi_{1132}(171,\cdot)\) \(\chi_{1132}(183,\cdot)\) \(\chi_{1132}(187,\cdot)\) \(\chi_{1132}(191,\cdot)\) \(\chi_{1132}(231,\cdot)\) \(\chi_{1132}(243,\cdot)\) \(\chi_{1132}(247,\cdot)\) \(\chi_{1132}(255,\cdot)\) \(\chi_{1132}(259,\cdot)\) \(\chi_{1132}(295,\cdot)\) \(\chi_{1132}(303,\cdot)\) \(\chi_{1132}(331,\cdot)\) \(\chi_{1132}(339,\cdot)\) \(\chi_{1132}(351,\cdot)\) \(\chi_{1132}(355,\cdot)\) \(\chi_{1132}(363,\cdot)\) \(\chi_{1132}(371,\cdot)\) \(\chi_{1132}(387,\cdot)\) ...

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: $\Q(\zeta_{141})$
Fixed field: Number field defined by a degree 282 polynomial (not computed)

Values on generators

\((567,569)\) → \((-1,e\left(\frac{215}{282}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1132 }(139, a) \) \(1\)\(1\)\(e\left(\frac{37}{141}\right)\)\(e\left(\frac{181}{282}\right)\)\(e\left(\frac{127}{282}\right)\)\(e\left(\frac{74}{141}\right)\)\(e\left(\frac{109}{282}\right)\)\(e\left(\frac{92}{141}\right)\)\(e\left(\frac{85}{94}\right)\)\(e\left(\frac{139}{282}\right)\)\(e\left(\frac{15}{47}\right)\)\(e\left(\frac{67}{94}\right)\)
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)
 
\( \chi_{ 1132 }(139,a) \;\) at \(\;a = \) e.g. 2