Properties

Label 1021.81
Modulus $1021$
Conductor $1021$
Order $17$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1021, base_ring=CyclotomicField(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([2]))
 
pari: [g,chi] = znchar(Mod(81,1021))
 

Basic properties

Modulus: \(1021\)
Conductor: \(1021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
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.j

\(\chi_{1021}(9,\cdot)\) \(\chi_{1021}(81,\cdot)\) \(\chi_{1021}(227,\cdot)\) \(\chi_{1021}(340,\cdot)\) \(\chi_{1021}(435,\cdot)\) \(\chi_{1021}(479,\cdot)\) \(\chi_{1021}(507,\cdot)\) \(\chi_{1021}(521,\cdot)\) \(\chi_{1021}(605,\cdot)\) \(\chi_{1021}(729,\cdot)\) \(\chi_{1021}(737,\cdot)\) \(\chi_{1021}(778,\cdot)\) \(\chi_{1021}(852,\cdot)\) \(\chi_{1021}(876,\cdot)\) \(\chi_{1021}(994,\cdot)\) \(\chi_{1021}(1018,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 17 polynomial

Values on generators

\(10\) → \(e\left(\frac{1}{17}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1021 }(81, a) \) \(1\)\(1\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{16}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1021 }(81,a) \;\) at \(\;a = \) e.g. 2