↑

Properties

Label 6727.3219
Modulus $6727$
Conductor $6727$
Order $186$
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(6727, base_ring=CyclotomicField(186)) M = H._module chi = DirichletCharacter(H, M([93,73]))
 
Copy content gp:[g,chi] = znchar(Mod(3219, 6727))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6727.3219");
 

Basic properties

Modulus: \(6727\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6727\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(186\)
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 6727.bz

\(\chi_{6727}(6,\cdot)\) \(\chi_{6727}(181,\cdot)\) \(\chi_{6727}(223,\cdot)\) \(\chi_{6727}(398,\cdot)\) \(\chi_{6727}(615,\cdot)\) \(\chi_{6727}(657,\cdot)\) \(\chi_{6727}(832,\cdot)\) \(\chi_{6727}(874,\cdot)\) \(\chi_{6727}(1049,\cdot)\) \(\chi_{6727}(1091,\cdot)\) \(\chi_{6727}(1266,\cdot)\) \(\chi_{6727}(1308,\cdot)\) \(\chi_{6727}(1525,\cdot)\) \(\chi_{6727}(1700,\cdot)\) \(\chi_{6727}(1742,\cdot)\) \(\chi_{6727}(1917,\cdot)\) \(\chi_{6727}(1959,\cdot)\) \(\chi_{6727}(2134,\cdot)\) \(\chi_{6727}(2176,\cdot)\) \(\chi_{6727}(2351,\cdot)\) \(\chi_{6727}(2393,\cdot)\) \(\chi_{6727}(2568,\cdot)\) \(\chi_{6727}(2610,\cdot)\) \(\chi_{6727}(2785,\cdot)\) \(\chi_{6727}(2827,\cdot)\) \(\chi_{6727}(3002,\cdot)\) \(\chi_{6727}(3044,\cdot)\) \(\chi_{6727}(3219,\cdot)\) \(\chi_{6727}(3261,\cdot)\) \(\chi_{6727}(3436,\cdot)\) ...

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: $\Q(\zeta_{93})$
Fixed field: Number field defined by a degree 186 polynomial (not computed)

Values on generators

\((962,5769)\) → \((-1,e\left(\frac{73}{186}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6727 }(3219, a) \) \(1\)\(1\)\(e\left(\frac{28}{31}\right)\)\(e\left(\frac{83}{93}\right)\)\(e\left(\frac{25}{31}\right)\)\(e\left(\frac{35}{186}\right)\)\(e\left(\frac{74}{93}\right)\)\(e\left(\frac{22}{31}\right)\)\(e\left(\frac{73}{93}\right)\)\(e\left(\frac{17}{186}\right)\)\(e\left(\frac{143}{186}\right)\)\(e\left(\frac{65}{93}\right)\)
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)
 
\( \chi_{ 6727 }(3219,a) \;\) at \(\;a = \) e.g. 2