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Properties

Label 2971.288
Modulus $2971$
Conductor $2971$
Order $990$
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(2971, base_ring=CyclotomicField(990)) M = H._module chi = DirichletCharacter(H, M([899]))
 
Copy content gp:[g,chi] = znchar(Mod(288, 2971))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2971.288");
 

Basic properties

Modulus: \(2971\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2971\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(990\)
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 2971.bd

\(\chi_{2971}(3,\cdot)\) \(\chi_{2971}(14,\cdot)\) \(\chi_{2971}(18,\cdot)\) \(\chi_{2971}(43,\cdot)\) \(\chi_{2971}(53,\cdot)\) \(\chi_{2971}(56,\cdot)\) \(\chi_{2971}(58,\cdot)\) \(\chi_{2971}(72,\cdot)\) \(\chi_{2971}(98,\cdot)\) \(\chi_{2971}(101,\cdot)\) \(\chi_{2971}(126,\cdot)\) \(\chi_{2971}(161,\cdot)\) \(\chi_{2971}(162,\cdot)\) \(\chi_{2971}(172,\cdot)\) \(\chi_{2971}(173,\cdot)\) \(\chi_{2971}(192,\cdot)\) \(\chi_{2971}(205,\cdot)\) \(\chi_{2971}(206,\cdot)\) \(\chi_{2971}(207,\cdot)\) \(\chi_{2971}(244,\cdot)\) \(\chi_{2971}(250,\cdot)\) \(\chi_{2971}(288,\cdot)\) \(\chi_{2971}(292,\cdot)\) \(\chi_{2971}(301,\cdot)\) \(\chi_{2971}(325,\cdot)\) \(\chi_{2971}(330,\cdot)\) \(\chi_{2971}(341,\cdot)\) \(\chi_{2971}(353,\cdot)\) \(\chi_{2971}(375,\cdot)\) \(\chi_{2971}(380,\cdot)\) ...

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: $\Q(\zeta_{495})$
Fixed field: Number field defined by a degree 990 polynomial (not computed)

Values on generators

\(10\) → \(e\left(\frac{899}{990}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2971 }(288, a) \) \(-1\)\(1\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{23}{330}\right)\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{58}{495}\right)\)\(e\left(\frac{142}{165}\right)\)\(e\left(\frac{68}{165}\right)\)\(e\left(\frac{41}{110}\right)\)\(e\left(\frac{23}{165}\right)\)\(e\left(\frac{899}{990}\right)\)\(e\left(\frac{379}{990}\right)\)
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)
 
\( \chi_{ 2971 }(288,a) \;\) at \(\;a = \) e.g. 2