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Properties

Label 5459.38
Modulus $5459$
Conductor $5459$
Order $1326$
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(5459, base_ring=CyclotomicField(1326)) M = H._module chi = DirichletCharacter(H, M([969,286]))
 
Copy content gp:[g,chi] = znchar(Mod(38, 5459))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5459.38");
 

Basic properties

Modulus: \(5459\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5459\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1326\)
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 5459.bs

\(\chi_{5459}(4,\cdot)\) \(\chi_{5459}(7,\cdot)\) \(\chi_{5459}(17,\cdot)\) \(\chi_{5459}(25,\cdot)\) \(\chi_{5459}(29,\cdot)\) \(\chi_{5459}(38,\cdot)\) \(\chi_{5459}(59,\cdot)\) \(\chi_{5459}(60,\cdot)\) \(\chi_{5459}(82,\cdot)\) \(\chi_{5459}(91,\cdot)\) \(\chi_{5459}(110,\cdot)\) \(\chi_{5459}(131,\cdot)\) \(\chi_{5459}(135,\cdot)\) \(\chi_{5459}(144,\cdot)\) \(\chi_{5459}(163,\cdot)\) \(\chi_{5459}(166,\cdot)\) \(\chi_{5459}(221,\cdot)\) \(\chi_{5459}(223,\cdot)\) \(\chi_{5459}(255,\cdot)\) \(\chi_{5459}(269,\cdot)\) \(\chi_{5459}(274,\cdot)\) \(\chi_{5459}(303,\cdot)\) \(\chi_{5459}(324,\cdot)\) \(\chi_{5459}(325,\cdot)\) \(\chi_{5459}(327,\cdot)\) \(\chi_{5459}(335,\cdot)\) \(\chi_{5459}(347,\cdot)\) \(\chi_{5459}(358,\cdot)\) \(\chi_{5459}(361,\cdot)\) \(\chi_{5459}(377,\cdot)\) ...

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: $\Q(\zeta_{663})$
Fixed field: Number field defined by a degree 1326 polynomial (not computed)

Values on generators

\((3606,1962)\) → \((e\left(\frac{19}{26}\right),e\left(\frac{11}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5459 }(38, a) \) \(1\)\(1\)\(e\left(\frac{293}{1326}\right)\)\(e\left(\frac{369}{442}\right)\)\(e\left(\frac{293}{663}\right)\)\(e\left(\frac{745}{1326}\right)\)\(e\left(\frac{37}{663}\right)\)\(e\left(\frac{62}{663}\right)\)\(e\left(\frac{293}{442}\right)\)\(e\left(\frac{148}{221}\right)\)\(e\left(\frac{173}{221}\right)\)\(e\left(\frac{359}{663}\right)\)
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)
 
\( \chi_{ 5459 }(38,a) \;\) at \(\;a = \) e.g. 2