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Properties

Label 87451.4924
Modulus $87451$
Conductor $87451$
Order $186$
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(87451, base_ring=CyclotomicField(186)) M = H._module chi = DirichletCharacter(H, M([31,155,91]))
 
Copy content gp:[g,chi] = znchar(Mod(4924, 87451))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87451.4924");
 

Basic properties

Modulus: \(87451\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(87451\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(186\)
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 87451.pi

\(\chi_{87451}(719,\cdot)\) \(\chi_{87451}(2103,\cdot)\) \(\chi_{87451}(3540,\cdot)\) \(\chi_{87451}(4924,\cdot)\) \(\chi_{87451}(6361,\cdot)\) \(\chi_{87451}(7745,\cdot)\) \(\chi_{87451}(9182,\cdot)\) \(\chi_{87451}(10566,\cdot)\) \(\chi_{87451}(12003,\cdot)\) \(\chi_{87451}(13387,\cdot)\) \(\chi_{87451}(14824,\cdot)\) \(\chi_{87451}(16208,\cdot)\) \(\chi_{87451}(17645,\cdot)\) \(\chi_{87451}(19029,\cdot)\) \(\chi_{87451}(20466,\cdot)\) \(\chi_{87451}(21850,\cdot)\) \(\chi_{87451}(23287,\cdot)\) \(\chi_{87451}(24671,\cdot)\) \(\chi_{87451}(26108,\cdot)\) \(\chi_{87451}(27492,\cdot)\) \(\chi_{87451}(28929,\cdot)\) \(\chi_{87451}(31750,\cdot)\) \(\chi_{87451}(33134,\cdot)\) \(\chi_{87451}(34571,\cdot)\) \(\chi_{87451}(35955,\cdot)\) \(\chi_{87451}(37392,\cdot)\) \(\chi_{87451}(38776,\cdot)\) \(\chi_{87451}(40213,\cdot)\) \(\chi_{87451}(41597,\cdot)\) \(\chi_{87451}(43034,\cdot)\) ...

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: $\Q(\zeta_{93})$
Fixed field: Number field defined by a degree 186 polynomial (not computed)

Values on generators

\((74959,73998,39404)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{91}{186}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 87451 }(4924, a) \) \(1\)\(1\)\(e\left(\frac{85}{186}\right)\)\(e\left(\frac{92}{93}\right)\)\(e\left(\frac{85}{93}\right)\)\(e\left(\frac{56}{93}\right)\)\(e\left(\frac{83}{186}\right)\)\(e\left(\frac{23}{62}\right)\)\(e\left(\frac{91}{93}\right)\)\(e\left(\frac{11}{186}\right)\)\(e\left(\frac{49}{93}\right)\)\(e\left(\frac{28}{31}\right)\)
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)
 
\( \chi_{ 87451 }(4924,a) \;\) at \(\;a = \) e.g. 2