Properties

Label 8011.20
Modulus $8011$
Conductor $8011$
Order $1335$
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(2670))
 
M = H._module
 
chi = DirichletCharacter(H, M([1298]))
 
pari: [g,chi] = znchar(Mod(20,8011))
 

Basic properties

Modulus: \(8011\)
Conductor: \(8011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1335\)
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.t

\(\chi_{8011}(4,\cdot)\) \(\chi_{8011}(9,\cdot)\) \(\chi_{8011}(16,\cdot)\) \(\chi_{8011}(20,\cdot)\) \(\chi_{8011}(24,\cdot)\) \(\chi_{8011}(33,\cdot)\) \(\chi_{8011}(45,\cdot)\) \(\chi_{8011}(54,\cdot)\) \(\chi_{8011}(67,\cdot)\) \(\chi_{8011}(71,\cdot)\) \(\chi_{8011}(81,\cdot)\) \(\chi_{8011}(91,\cdot)\) \(\chi_{8011}(96,\cdot)\) \(\chi_{8011}(100,\cdot)\) \(\chi_{8011}(110,\cdot)\) \(\chi_{8011}(116,\cdot)\) \(\chi_{8011}(120,\cdot)\) \(\chi_{8011}(133,\cdot)\) \(\chi_{8011}(141,\cdot)\) \(\chi_{8011}(144,\cdot)\) \(\chi_{8011}(165,\cdot)\) \(\chi_{8011}(198,\cdot)\) \(\chi_{8011}(241,\cdot)\) \(\chi_{8011}(256,\cdot)\) \(\chi_{8011}(257,\cdot)\) \(\chi_{8011}(261,\cdot)\) \(\chi_{8011}(270,\cdot)\) \(\chi_{8011}(277,\cdot)\) \(\chi_{8011}(279,\cdot)\) \(\chi_{8011}(284,\cdot)\) ...

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: $\Q(\zeta_{1335})$
Fixed field: Number field defined by a degree 1335 polynomial (not computed)

Values on generators

\(14\) → \(e\left(\frac{649}{1335}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8011 }(20, a) \) \(1\)\(1\)\(e\left(\frac{344}{445}\right)\)\(e\left(\frac{386}{445}\right)\)\(e\left(\frac{243}{445}\right)\)\(e\left(\frac{219}{445}\right)\)\(e\left(\frac{57}{89}\right)\)\(e\left(\frac{952}{1335}\right)\)\(e\left(\frac{142}{445}\right)\)\(e\left(\frac{327}{445}\right)\)\(e\left(\frac{118}{445}\right)\)\(e\left(\frac{51}{445}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8011 }(20,a) \;\) at \(\;a = \) e.g. 2