↑

Properties

Label 2767.1020
Modulus $2767$
Conductor $2767$
Order $461$
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(2767, base_ring=CyclotomicField(922)) M = H._module chi = DirichletCharacter(H, M([328]))
 
Copy content gp:[g,chi] = znchar(Mod(1020, 2767))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2767.1020");
 

Basic properties

Modulus: \(2767\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2767\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(461\)
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 2767.e

\(\chi_{2767}(2,\cdot)\) \(\chi_{2767}(4,\cdot)\) \(\chi_{2767}(8,\cdot)\) \(\chi_{2767}(11,\cdot)\) \(\chi_{2767}(15,\cdot)\) \(\chi_{2767}(16,\cdot)\) \(\chi_{2767}(17,\cdot)\) \(\chi_{2767}(19,\cdot)\) \(\chi_{2767}(22,\cdot)\) \(\chi_{2767}(23,\cdot)\) \(\chi_{2767}(30,\cdot)\) \(\chi_{2767}(32,\cdot)\) \(\chi_{2767}(34,\cdot)\) \(\chi_{2767}(38,\cdot)\) \(\chi_{2767}(44,\cdot)\) \(\chi_{2767}(46,\cdot)\) \(\chi_{2767}(49,\cdot)\) \(\chi_{2767}(60,\cdot)\) \(\chi_{2767}(64,\cdot)\) \(\chi_{2767}(65,\cdot)\) \(\chi_{2767}(68,\cdot)\) \(\chi_{2767}(76,\cdot)\) \(\chi_{2767}(87,\cdot)\) \(\chi_{2767}(88,\cdot)\) \(\chi_{2767}(92,\cdot)\) \(\chi_{2767}(93,\cdot)\) \(\chi_{2767}(98,\cdot)\) \(\chi_{2767}(103,\cdot)\) \(\chi_{2767}(120,\cdot)\) \(\chi_{2767}(121,\cdot)\) ...

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: $\Q(\zeta_{461})$
Fixed field: Number field defined by a degree 461 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{164}{461}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2767 }(1020, a) \) \(1\)\(1\)\(e\left(\frac{171}{461}\right)\)\(e\left(\frac{164}{461}\right)\)\(e\left(\frac{342}{461}\right)\)\(e\left(\frac{21}{461}\right)\)\(e\left(\frac{335}{461}\right)\)\(e\left(\frac{86}{461}\right)\)\(e\left(\frac{52}{461}\right)\)\(e\left(\frac{328}{461}\right)\)\(e\left(\frac{192}{461}\right)\)\(e\left(\frac{187}{461}\right)\)
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)
 
\( \chi_{ 2767 }(1020,a) \;\) at \(\;a = \) e.g. 2