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Properties

Label 5753.1055
Modulus $5753$
Conductor $5753$
Order $58$
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(5753, base_ring=CyclotomicField(58)) M = H._module chi = DirichletCharacter(H, M([29,8]))
 
Copy content gp:[g,chi] = znchar(Mod(1055, 5753))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5753.1055");
 

Basic properties

Modulus: \(5753\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5753\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(58\)
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 5753.x

\(\chi_{5753}(43,\cdot)\) \(\chi_{5753}(285,\cdot)\) \(\chi_{5753}(428,\cdot)\) \(\chi_{5753}(604,\cdot)\) \(\chi_{5753}(714,\cdot)\) \(\chi_{5753}(868,\cdot)\) \(\chi_{5753}(1055,\cdot)\) \(\chi_{5753}(1220,\cdot)\) \(\chi_{5753}(1440,\cdot)\) \(\chi_{5753}(2034,\cdot)\) \(\chi_{5753}(2089,\cdot)\) \(\chi_{5753}(2298,\cdot)\) \(\chi_{5753}(2749,\cdot)\) \(\chi_{5753}(3002,\cdot)\) \(\chi_{5753}(3211,\cdot)\) \(\chi_{5753}(3288,\cdot)\) \(\chi_{5753}(3442,\cdot)\) \(\chi_{5753}(4069,\cdot)\) \(\chi_{5753}(4157,\cdot)\) \(\chi_{5753}(4344,\cdot)\) \(\chi_{5753}(4410,\cdot)\) \(\chi_{5753}(4674,\cdot)\) \(\chi_{5753}(4718,\cdot)\) \(\chi_{5753}(4806,\cdot)\) \(\chi_{5753}(4828,\cdot)\) \(\chi_{5753}(5180,\cdot)\) \(\chi_{5753}(5510,\cdot)\) \(\chi_{5753}(5598,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((1047,4709)\) → \((-1,e\left(\frac{4}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 5753 }(1055, a) \) \(-1\)\(1\)\(e\left(\frac{37}{58}\right)\)\(e\left(\frac{28}{29}\right)\)\(e\left(\frac{8}{29}\right)\)\(e\left(\frac{22}{29}\right)\)\(e\left(\frac{35}{58}\right)\)\(e\left(\frac{9}{58}\right)\)\(e\left(\frac{53}{58}\right)\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{23}{58}\right)\)\(e\left(\frac{7}{29}\right)\)
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)
 
\( \chi_{ 5753 }(1055,a) \;\) at \(\;a = \) e.g. 2