Properties

Label 8011.22
Modulus $8011$
Conductor $8011$
Order $267$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8011, base_ring=CyclotomicField(534))
 
M = H._module
 
chi = DirichletCharacter(H, M([116]))
 
pari: [g,chi] = znchar(Mod(22,8011))
 

Basic properties

Modulus: \(8011\)
Conductor: \(8011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(267\)
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 8011.o

\(\chi_{8011}(22,\cdot)\) \(\chi_{8011}(80,\cdot)\) \(\chi_{8011}(94,\cdot)\) \(\chi_{8011}(124,\cdot)\) \(\chi_{8011}(132,\cdot)\) \(\chi_{8011}(225,\cdot)\) \(\chi_{8011}(355,\cdot)\) \(\chi_{8011}(401,\cdot)\) \(\chi_{8011}(480,\cdot)\) \(\chi_{8011}(484,\cdot)\) \(\chi_{8011}(500,\cdot)\) \(\chi_{8011}(511,\cdot)\) \(\chi_{8011}(534,\cdot)\) \(\chi_{8011}(564,\cdot)\) \(\chi_{8011}(581,\cdot)\) \(\chi_{8011}(744,\cdot)\) \(\chi_{8011}(751,\cdot)\) \(\chi_{8011}(777,\cdot)\) \(\chi_{8011}(792,\cdot)\) \(\chi_{8011}(825,\cdot)\) \(\chi_{8011}(904,\cdot)\) \(\chi_{8011}(939,\cdot)\) \(\chi_{8011}(1024,\cdot)\) \(\chi_{8011}(1028,\cdot)\) \(\chi_{8011}(1072,\cdot)\) \(\chi_{8011}(1099,\cdot)\) \(\chi_{8011}(1191,\cdot)\) \(\chi_{8011}(1258,\cdot)\) \(\chi_{8011}(1350,\cdot)\) \(\chi_{8011}(1402,\cdot)\) ...

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: $\Q(\zeta_{267})$
Fixed field: Number field defined by a degree 267 polynomial (not computed)

Values on generators

\(14\) → \(e\left(\frac{58}{267}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8011 }(22, a) \) \(1\)\(1\)\(e\left(\frac{28}{89}\right)\)\(e\left(\frac{19}{89}\right)\)\(e\left(\frac{56}{89}\right)\)\(e\left(\frac{23}{89}\right)\)\(e\left(\frac{47}{89}\right)\)\(e\left(\frac{241}{267}\right)\)\(e\left(\frac{84}{89}\right)\)\(e\left(\frac{38}{89}\right)\)\(e\left(\frac{51}{89}\right)\)\(e\left(\frac{59}{89}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8011 }(22,a) \;\) at \(\;a = \) e.g. 2