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Properties

Label 5082.65
Modulus $5082$
Conductor $2541$
Order $66$
Real no
Primitive no
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(5082, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([33,22,39]))
 
Copy content gp:[g,chi] = znchar(Mod(65, 5082))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5082.65");
 

Basic properties

Modulus: \(5082\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2541\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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: no, induced from \(\chi_{2541}(65,\cdot)\)
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 5082.bt

\(\chi_{5082}(65,\cdot)\) \(\chi_{5082}(263,\cdot)\) \(\chi_{5082}(527,\cdot)\) \(\chi_{5082}(989,\cdot)\) \(\chi_{5082}(1187,\cdot)\) \(\chi_{5082}(1649,\cdot)\) \(\chi_{5082}(1913,\cdot)\) \(\chi_{5082}(2111,\cdot)\) \(\chi_{5082}(2375,\cdot)\) \(\chi_{5082}(2573,\cdot)\) \(\chi_{5082}(2837,\cdot)\) \(\chi_{5082}(3035,\cdot)\) \(\chi_{5082}(3299,\cdot)\) \(\chi_{5082}(3497,\cdot)\) \(\chi_{5082}(3761,\cdot)\) \(\chi_{5082}(3959,\cdot)\) \(\chi_{5082}(4223,\cdot)\) \(\chi_{5082}(4421,\cdot)\) \(\chi_{5082}(4685,\cdot)\) \(\chi_{5082}(4883,\cdot)\)

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: \(\Q(\zeta_{33})\)
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 66 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((3389,4357,2059)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{13}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5082 }(65, a) \) \(1\)\(1\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{1}{11}\right)\)
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)
 
\( \chi_{ 5082 }(65,a) \;\) at \(\;a = \) e.g. 2