Properties

Label 1021.87
Modulus $1021$
Conductor $1021$
Order $255$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1021, base_ring=CyclotomicField(510))
 
M = H._module
 
chi = DirichletCharacter(H, M([226]))
 
pari: [g,chi] = znchar(Mod(87,1021))
 

Basic properties

Modulus: \(1021\)
Conductor: \(1021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(255\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1021.u

\(\chi_{1021}(5,\cdot)\) \(\chi_{1021}(11,\cdot)\) \(\chi_{1021}(23,\cdot)\) \(\chi_{1021}(25,\cdot)\) \(\chi_{1021}(45,\cdot)\) \(\chi_{1021}(47,\cdot)\) \(\chi_{1021}(51,\cdot)\) \(\chi_{1021}(55,\cdot)\) \(\chi_{1021}(60,\cdot)\) \(\chi_{1021}(68,\cdot)\) \(\chi_{1021}(70,\cdot)\) \(\chi_{1021}(87,\cdot)\) \(\chi_{1021}(99,\cdot)\) \(\chi_{1021}(114,\cdot)\) \(\chi_{1021}(116,\cdot)\) \(\chi_{1021}(121,\cdot)\) \(\chi_{1021}(133,\cdot)\) \(\chi_{1021}(154,\cdot)\) \(\chi_{1021}(164,\cdot)\) \(\chi_{1021}(167,\cdot)\) \(\chi_{1021}(176,\cdot)\) \(\chi_{1021}(183,\cdot)\) \(\chi_{1021}(193,\cdot)\) \(\chi_{1021}(197,\cdot)\) \(\chi_{1021}(206,\cdot)\) \(\chi_{1021}(207,\cdot)\) \(\chi_{1021}(222,\cdot)\) \(\chi_{1021}(225,\cdot)\) \(\chi_{1021}(233,\cdot)\) \(\chi_{1021}(244,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{255})$
Fixed field: Number field defined by a degree 255 polynomial (not computed)

Values on generators

\(10\) → \(e\left(\frac{113}{255}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1021 }(87, a) \) \(1\)\(1\)\(e\left(\frac{48}{85}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{11}{85}\right)\)\(e\left(\frac{224}{255}\right)\)\(e\left(\frac{18}{85}\right)\)\(e\left(\frac{58}{85}\right)\)\(e\left(\frac{59}{85}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{113}{255}\right)\)\(e\left(\frac{23}{255}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1021 }(87,a) \;\) at \(\;a = \) e.g. 2