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Properties

Label 3671.6
Modulus $3671$
Conductor $3671$
Order $1835$
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(3671, base_ring=CyclotomicField(3670)) M = H._module chi = DirichletCharacter(H, M([3138]))
 
Copy content gp:[g,chi] = znchar(Mod(6, 3671))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3671.6");
 

Basic properties

Modulus: \(3671\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3671\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1835\)
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 3671.g

\(\chi_{3671}(2,\cdot)\) \(\chi_{3671}(3,\cdot)\) \(\chi_{3671}(4,\cdot)\) \(\chi_{3671}(5,\cdot)\) \(\chi_{3671}(6,\cdot)\) \(\chi_{3671}(7,\cdot)\) \(\chi_{3671}(8,\cdot)\) \(\chi_{3671}(9,\cdot)\) \(\chi_{3671}(11,\cdot)\) \(\chi_{3671}(12,\cdot)\) \(\chi_{3671}(16,\cdot)\) \(\chi_{3671}(17,\cdot)\) \(\chi_{3671}(18,\cdot)\) \(\chi_{3671}(20,\cdot)\) \(\chi_{3671}(23,\cdot)\) \(\chi_{3671}(24,\cdot)\) \(\chi_{3671}(25,\cdot)\) \(\chi_{3671}(27,\cdot)\) \(\chi_{3671}(28,\cdot)\) \(\chi_{3671}(30,\cdot)\) \(\chi_{3671}(35,\cdot)\) \(\chi_{3671}(36,\cdot)\) \(\chi_{3671}(40,\cdot)\) \(\chi_{3671}(42,\cdot)\) \(\chi_{3671}(44,\cdot)\) \(\chi_{3671}(45,\cdot)\) \(\chi_{3671}(46,\cdot)\) \(\chi_{3671}(47,\cdot)\) \(\chi_{3671}(49,\cdot)\) \(\chi_{3671}(50,\cdot)\) ...

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: $\Q(\zeta_{1835})$
Fixed field: Number field defined by a degree 1835 polynomial (not computed)

Values on generators

\(13\) → \(e\left(\frac{1569}{1835}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3671 }(6, a) \) \(1\)\(1\)\(e\left(\frac{976}{1835}\right)\)\(e\left(\frac{1076}{1835}\right)\)\(e\left(\frac{117}{1835}\right)\)\(e\left(\frac{1594}{1835}\right)\)\(e\left(\frac{217}{1835}\right)\)\(e\left(\frac{679}{1835}\right)\)\(e\left(\frac{1093}{1835}\right)\)\(e\left(\frac{317}{1835}\right)\)\(e\left(\frac{147}{367}\right)\)\(e\left(\frac{1199}{1835}\right)\)
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)
 
\( \chi_{ 3671 }(6,a) \;\) at \(\;a = \) e.g. 2