↑

Properties

Label 1267.83
Modulus $1267$
Conductor $1267$
Order $180$
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(1267, base_ring=CyclotomicField(180)) M = H._module chi = DirichletCharacter(H, M([90,121]))
 
Copy content gp:[g,chi] = znchar(Mod(83, 1267))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1267.83");
 

Basic properties

Modulus: \(1267\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1267\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(180\)
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 1267.do

\(\chi_{1267}(41,\cdot)\) \(\chi_{1267}(69,\cdot)\) \(\chi_{1267}(76,\cdot)\) \(\chi_{1267}(83,\cdot)\) \(\chi_{1267}(90,\cdot)\) \(\chi_{1267}(97,\cdot)\) \(\chi_{1267}(104,\cdot)\) \(\chi_{1267}(118,\cdot)\) \(\chi_{1267}(153,\cdot)\) \(\chi_{1267}(160,\cdot)\) \(\chi_{1267}(202,\cdot)\) \(\chi_{1267}(209,\cdot)\) \(\chi_{1267}(244,\cdot)\) \(\chi_{1267}(258,\cdot)\) \(\chi_{1267}(265,\cdot)\) \(\chi_{1267}(272,\cdot)\) \(\chi_{1267}(279,\cdot)\) \(\chi_{1267}(286,\cdot)\) \(\chi_{1267}(293,\cdot)\) \(\chi_{1267}(321,\cdot)\) \(\chi_{1267}(412,\cdot)\) \(\chi_{1267}(419,\cdot)\) \(\chi_{1267}(440,\cdot)\) \(\chi_{1267}(447,\cdot)\) \(\chi_{1267}(489,\cdot)\) \(\chi_{1267}(496,\cdot)\) \(\chi_{1267}(545,\cdot)\) \(\chi_{1267}(566,\cdot)\) \(\chi_{1267}(601,\cdot)\) \(\chi_{1267}(671,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((906,183)\) → \((-1,e\left(\frac{121}{180}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1267 }(83, a) \) \(1\)\(1\)\(e\left(\frac{121}{180}\right)\)\(e\left(\frac{13}{90}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{7}{180}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{22}{45}\right)\)
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)
 
\( \chi_{ 1267 }(83,a) \;\) at \(\;a = \) e.g. 2