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Properties

Label 5881.847
Modulus $5881$
Conductor $5881$
Order $490$
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(5881, base_ring=CyclotomicField(490)) M = H._module chi = DirichletCharacter(H, M([411]))
 
Copy content gp:[g,chi] = znchar(Mod(847, 5881))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5881.847");
 

Basic properties

Modulus: \(5881\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5881\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(490\)
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 5881.bm

\(\chi_{5881}(8,\cdot)\) \(\chi_{5881}(15,\cdot)\) \(\chi_{5881}(29,\cdot)\) \(\chi_{5881}(36,\cdot)\) \(\chi_{5881}(49,\cdot)\) \(\chi_{5881}(73,\cdot)\) \(\chi_{5881}(162,\cdot)\) \(\chi_{5881}(183,\cdot)\) \(\chi_{5881}(189,\cdot)\) \(\chi_{5881}(205,\cdot)\) \(\chi_{5881}(230,\cdot)\) \(\chi_{5881}(289,\cdot)\) \(\chi_{5881}(334,\cdot)\) \(\chi_{5881}(340,\cdot)\) \(\chi_{5881}(485,\cdot)\) \(\chi_{5881}(512,\cdot)\) \(\chi_{5881}(552,\cdot)\) \(\chi_{5881}(627,\cdot)\) \(\chi_{5881}(729,\cdot)\) \(\chi_{5881}(750,\cdot)\) \(\chi_{5881}(781,\cdot)\) \(\chi_{5881}(827,\cdot)\) \(\chi_{5881}(843,\cdot)\) \(\chi_{5881}(847,\cdot)\) \(\chi_{5881}(853,\cdot)\) \(\chi_{5881}(865,\cdot)\) \(\chi_{5881}(871,\cdot)\) \(\chi_{5881}(872,\cdot)\) \(\chi_{5881}(898,\cdot)\) \(\chi_{5881}(952,\cdot)\) ...

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: $\Q(\zeta_{245})$
Fixed field: Number field defined by a degree 490 polynomial (not computed)

Values on generators

\(31\) → \(e\left(\frac{411}{490}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5881 }(847, a) \) \(1\)\(1\)\(e\left(\frac{89}{245}\right)\)\(e\left(\frac{192}{245}\right)\)\(e\left(\frac{178}{245}\right)\)\(e\left(\frac{82}{245}\right)\)\(e\left(\frac{36}{245}\right)\)\(e\left(\frac{97}{245}\right)\)\(e\left(\frac{22}{245}\right)\)\(e\left(\frac{139}{245}\right)\)\(e\left(\frac{171}{245}\right)\)\(e\left(\frac{109}{490}\right)\)
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)
 
\( \chi_{ 5881 }(847,a) \;\) at \(\;a = \) e.g. 2