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Properties

Label 1255.1033
Modulus $1255$
Conductor $1255$
Order $500$
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(1255, base_ring=CyclotomicField(500)) M = H._module chi = DirichletCharacter(H, M([375,166]))
 
Copy content gp:[g,chi] = znchar(Mod(1033, 1255))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1255.1033");
 

Basic properties

Modulus: \(1255\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1255\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(500\)
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 1255.x

\(\chi_{1255}(18,\cdot)\) \(\chi_{1255}(33,\cdot)\) \(\chi_{1255}(37,\cdot)\) \(\chi_{1255}(42,\cdot)\) \(\chi_{1255}(43,\cdot)\) \(\chi_{1255}(53,\cdot)\) \(\chi_{1255}(57,\cdot)\) \(\chi_{1255}(62,\cdot)\) \(\chi_{1255}(72,\cdot)\) \(\chi_{1255}(77,\cdot)\) \(\chi_{1255}(78,\cdot)\) \(\chi_{1255}(82,\cdot)\) \(\chi_{1255}(87,\cdot)\) \(\chi_{1255}(97,\cdot)\) \(\chi_{1255}(98,\cdot)\) \(\chi_{1255}(107,\cdot)\) \(\chi_{1255}(127,\cdot)\) \(\chi_{1255}(132,\cdot)\) \(\chi_{1255}(133,\cdot)\) \(\chi_{1255}(137,\cdot)\) \(\chi_{1255}(143,\cdot)\) \(\chi_{1255}(148,\cdot)\) \(\chi_{1255}(158,\cdot)\) \(\chi_{1255}(162,\cdot)\) \(\chi_{1255}(163,\cdot)\) \(\chi_{1255}(167,\cdot)\) \(\chi_{1255}(168,\cdot)\) \(\chi_{1255}(172,\cdot)\) \(\chi_{1255}(177,\cdot)\) \(\chi_{1255}(178,\cdot)\) ...

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: $\Q(\zeta_{500})$
Fixed field: Number field defined by a degree 500 polynomial (not computed)

Values on generators

\((252,6)\) → \((-i,e\left(\frac{83}{250}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1255 }(1033, a) \) \(1\)\(1\)\(e\left(\frac{77}{100}\right)\)\(e\left(\frac{281}{500}\right)\)\(e\left(\frac{27}{50}\right)\)\(e\left(\frac{83}{250}\right)\)\(e\left(\frac{43}{500}\right)\)\(e\left(\frac{31}{100}\right)\)\(e\left(\frac{31}{250}\right)\)\(e\left(\frac{13}{250}\right)\)\(e\left(\frac{51}{500}\right)\)\(e\left(\frac{337}{500}\right)\)
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)
 
\( \chi_{ 1255 }(1033,a) \;\) at \(\;a = \) e.g. 2