↑

Properties

Label 3671.1047
Modulus $3671$
Conductor $3671$
Order $367$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

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(734)) M = H._module chi = DirichletCharacter(H, M([416]))
 
Copy content gp:[g,chi] = znchar(Mod(1047, 3671))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3671.1047");
 

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: \(367\)
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.e

\(\chi_{3671}(10,\cdot)\) \(\chi_{3671}(14,\cdot)\) \(\chi_{3671}(15,\cdot)\) \(\chi_{3671}(21,\cdot)\) \(\chi_{3671}(22,\cdot)\) \(\chi_{3671}(31,\cdot)\) \(\chi_{3671}(32,\cdot)\) \(\chi_{3671}(33,\cdot)\) \(\chi_{3671}(34,\cdot)\) \(\chi_{3671}(48,\cdot)\) \(\chi_{3671}(51,\cdot)\) \(\chi_{3671}(72,\cdot)\) \(\chi_{3671}(83,\cdot)\) \(\chi_{3671}(92,\cdot)\) \(\chi_{3671}(100,\cdot)\) \(\chi_{3671}(108,\cdot)\) \(\chi_{3671}(138,\cdot)\) \(\chi_{3671}(140,\cdot)\) \(\chi_{3671}(150,\cdot)\) \(\chi_{3671}(162,\cdot)\) \(\chi_{3671}(188,\cdot)\) \(\chi_{3671}(191,\cdot)\) \(\chi_{3671}(196,\cdot)\) \(\chi_{3671}(207,\cdot)\) \(\chi_{3671}(210,\cdot)\) \(\chi_{3671}(220,\cdot)\) \(\chi_{3671}(225,\cdot)\) \(\chi_{3671}(236,\cdot)\) \(\chi_{3671}(243,\cdot)\) \(\chi_{3671}(247,\cdot)\) ...

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: $\Q(\zeta_{367})$
Fixed field: Number field defined by a degree 367 polynomial (not computed)

Values on generators

\(13\) → \(e\left(\frac{208}{367}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3671 }(1047, a) \) \(1\)\(1\)\(e\left(\frac{26}{367}\right)\)\(e\left(\frac{152}{367}\right)\)\(e\left(\frac{52}{367}\right)\)\(e\left(\frac{56}{367}\right)\)\(e\left(\frac{178}{367}\right)\)\(e\left(\frac{261}{367}\right)\)\(e\left(\frac{78}{367}\right)\)\(e\left(\frac{304}{367}\right)\)\(e\left(\frac{82}{367}\right)\)\(e\left(\frac{329}{367}\right)\)
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)
 
\( \chi_{ 3671 }(1047,a) \;\) at \(\;a = \) e.g. 2