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Properties

Label 5200.37
Modulus $5200$
Conductor $5200$
Order $60$
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(5200, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([0,15,27,35]))
 
Copy content gp:[g,chi] = znchar(Mod(37, 5200))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5200.37");
 

Basic properties

Modulus: \(5200\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5200\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(60\)
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 5200.jj

\(\chi_{5200}(37,\cdot)\) \(\chi_{5200}(253,\cdot)\) \(\chi_{5200}(917,\cdot)\) \(\chi_{5200}(1077,\cdot)\) \(\chi_{5200}(1133,\cdot)\) \(\chi_{5200}(2117,\cdot)\) \(\chi_{5200}(2173,\cdot)\) \(\chi_{5200}(2333,\cdot)\) \(\chi_{5200}(2997,\cdot)\) \(\chi_{5200}(3213,\cdot)\) \(\chi_{5200}(3373,\cdot)\) \(\chi_{5200}(4037,\cdot)\) \(\chi_{5200}(4197,\cdot)\) \(\chi_{5200}(4253,\cdot)\) \(\chi_{5200}(4413,\cdot)\) \(\chi_{5200}(5077,\cdot)\)

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: \(\Q(\zeta_{60})\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 60 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((1951,1301,4577,1601)\) → \((1,i,e\left(\frac{9}{20}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 5200 }(37, a) \) \(1\)\(1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{59}{60}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 5200 }(37,a) \;\) at \(\;a = \) e.g. 2