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Properties

Label 892607.293
Modulus $892607$
Conductor $892607$
Order $212366$
Real no
Primitive yes
Minimal yes
Parity odd

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(892607, base_ring=CyclotomicField(212366)) M = H._module chi = DirichletCharacter(H, M([67571,89419]))
 
Copy content gp:[g,chi] = znchar(Mod(293, 892607))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("892607.293");
 

Basic properties

Modulus: \(892607\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(892607\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(212366\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 892607.cr

\(\chi_{892607}(7,\cdot)\) \(\chi_{892607}(10,\cdot)\) \(\chi_{892607}(15,\cdot)\) \(\chi_{892607}(43,\cdot)\) \(\chi_{892607}(65,\cdot)\) \(\chi_{892607}(97,\cdot)\) \(\chi_{892607}(107,\cdot)\) \(\chi_{892607}(109,\cdot)\) \(\chi_{892607}(112,\cdot)\) \(\chi_{892607}(134,\cdot)\) \(\chi_{892607}(136,\cdot)\) \(\chi_{892607}(148,\cdot)\) \(\chi_{892607}(155,\cdot)\) \(\chi_{892607}(157,\cdot)\) \(\chi_{892607}(168,\cdot)\) \(\chi_{892607}(181,\cdot)\) \(\chi_{892607}(201,\cdot)\) \(\chi_{892607}(204,\cdot)\) \(\chi_{892607}(212,\cdot)\) \(\chi_{892607}(222,\cdot)\) \(\chi_{892607}(240,\cdot)\) \(\chi_{892607}(244,\cdot)\) \(\chi_{892607}(293,\cdot)\) \(\chi_{892607}(304,\cdot)\) \(\chi_{892607}(306,\cdot)\) \(\chi_{892607}(309,\cdot)\) \(\chi_{892607}(313,\cdot)\) \(\chi_{892607}(333,\cdot)\) \(\chi_{892607}(341,\cdot)\) \(\chi_{892607}(343,\cdot)\) ...

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: $\Q(\zeta_{106183})$
Fixed field: Number field defined by a degree 212366 polynomial (not computed)

Values on generators

\((465709,776182)\) → \((e\left(\frac{7}{22}\right),e\left(\frac{8129}{19306}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 892607 }(293, a) \) \(-1\)\(1\)\(e\left(\frac{12195}{212366}\right)\)\(e\left(\frac{189377}{212366}\right)\)\(e\left(\frac{12195}{106183}\right)\)\(e\left(\frac{31338}{106183}\right)\)\(e\left(\frac{14398}{15169}\right)\)\(e\left(\frac{106189}{212366}\right)\)\(e\left(\frac{36585}{212366}\right)\)\(e\left(\frac{83194}{106183}\right)\)\(e\left(\frac{74871}{212366}\right)\)\(e\left(\frac{9517}{106183}\right)\)
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)
 
\( \chi_{ 892607 }(293,a) \;\) at \(\;a = \) e.g. 2