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Properties

Label 5209.1009
Modulus $5209$
Conductor $5209$
Order $868$
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(5209, base_ring=CyclotomicField(868)) M = H._module chi = DirichletCharacter(H, M([67]))
 
Copy content gp:[g,chi] = znchar(Mod(1009, 5209))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5209.1009");
 

Basic properties

Modulus: \(5209\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5209\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(868\)
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 5209.bb

\(\chi_{5209}(15,\cdot)\) \(\chi_{5209}(30,\cdot)\) \(\chi_{5209}(60,\cdot)\) \(\chi_{5209}(63,\cdot)\) \(\chi_{5209}(125,\cdot)\) \(\chi_{5209}(129,\cdot)\) \(\chi_{5209}(130,\cdot)\) \(\chi_{5209}(240,\cdot)\) \(\chi_{5209}(250,\cdot)\) \(\chi_{5209}(252,\cdot)\) \(\chi_{5209}(258,\cdot)\) \(\chi_{5209}(260,\cdot)\) \(\chi_{5209}(273,\cdot)\) \(\chi_{5209}(279,\cdot)\) \(\chi_{5209}(287,\cdot)\) \(\chi_{5209}(343,\cdot)\) \(\chi_{5209}(347,\cdot)\) \(\chi_{5209}(363,\cdot)\) \(\chi_{5209}(373,\cdot)\) \(\chi_{5209}(405,\cdot)\) \(\chi_{5209}(417,\cdot)\) \(\chi_{5209}(480,\cdot)\) \(\chi_{5209}(500,\cdot)\) \(\chi_{5209}(516,\cdot)\) \(\chi_{5209}(517,\cdot)\) \(\chi_{5209}(520,\cdot)\) \(\chi_{5209}(525,\cdot)\) \(\chi_{5209}(534,\cdot)\) \(\chi_{5209}(561,\cdot)\) \(\chi_{5209}(574,\cdot)\) ...

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: $\Q(\zeta_{868})$
Fixed field: Number field defined by a degree 868 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{67}{868}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 5209 }(1009, a) \) \(1\)\(1\)\(e\left(\frac{110}{217}\right)\)\(e\left(\frac{25}{217}\right)\)\(e\left(\frac{3}{217}\right)\)\(e\left(\frac{109}{434}\right)\)\(e\left(\frac{135}{217}\right)\)\(e\left(\frac{387}{434}\right)\)\(e\left(\frac{113}{217}\right)\)\(e\left(\frac{50}{217}\right)\)\(e\left(\frac{47}{62}\right)\)\(e\left(\frac{45}{124}\right)\)
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)
 
\( \chi_{ 5209 }(1009,a) \;\) at \(\;a = \) e.g. 2