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Properties

Label 5060.2559
Modulus $5060$
Conductor $5060$
Order $110$
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(5060, base_ring=CyclotomicField(110)) M = H._module chi = DirichletCharacter(H, M([55,55,77,90]))
 
Copy content gp:[g,chi] = znchar(Mod(2559, 5060))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5060.2559");
 

Basic properties

Modulus: \(5060\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5060\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(110\)
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 5060.dg

\(\chi_{5060}(39,\cdot)\) \(\chi_{5060}(239,\cdot)\) \(\chi_{5060}(259,\cdot)\) \(\chi_{5060}(519,\cdot)\) \(\chi_{5060}(579,\cdot)\) \(\chi_{5060}(679,\cdot)\) \(\chi_{5060}(699,\cdot)\) \(\chi_{5060}(739,\cdot)\) \(\chi_{5060}(899,\cdot)\) \(\chi_{5060}(959,\cdot)\) \(\chi_{5060}(1139,\cdot)\) \(\chi_{5060}(1179,\cdot)\) \(\chi_{5060}(1359,\cdot)\) \(\chi_{5060}(1559,\cdot)\) \(\chi_{5060}(1619,\cdot)\) \(\chi_{5060}(1779,\cdot)\) \(\chi_{5060}(1899,\cdot)\) \(\chi_{5060}(2019,\cdot)\) \(\chi_{5060}(2059,\cdot)\) \(\chi_{5060}(2119,\cdot)\) \(\chi_{5060}(2239,\cdot)\) \(\chi_{5060}(2279,\cdot)\) \(\chi_{5060}(2339,\cdot)\) \(\chi_{5060}(2559,\cdot)\) \(\chi_{5060}(2879,\cdot)\) \(\chi_{5060}(2939,\cdot)\) \(\chi_{5060}(2999,\cdot)\) \(\chi_{5060}(3159,\cdot)\) \(\chi_{5060}(3339,\cdot)\) \(\chi_{5060}(3439,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((2531,3037,2301,3961)\) → \((-1,-1,e\left(\frac{7}{10}\right),e\left(\frac{9}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)\(31\)
\( \chi_{ 5060 }(2559, a) \) \(1\)\(1\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{49}{110}\right)\)\(e\left(\frac{21}{55}\right)\)\(e\left(\frac{36}{55}\right)\)\(e\left(\frac{29}{55}\right)\)\(e\left(\frac{48}{55}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{69}{110}\right)\)\(e\left(\frac{67}{110}\right)\)
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)
 
\( \chi_{ 5060 }(2559,a) \;\) at \(\;a = \) e.g. 2