Properties

Label 8712.139
Modulus $8712$
Conductor $8712$
Order $330$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8712, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,165,110,201]))
 
pari: [g,chi] = znchar(Mod(139,8712))
 

Basic properties

Modulus: \(8712\)
Conductor: \(8712\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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 8712.eq

\(\chi_{8712}(139,\cdot)\) \(\chi_{8712}(211,\cdot)\) \(\chi_{8712}(259,\cdot)\) \(\chi_{8712}(283,\cdot)\) \(\chi_{8712}(547,\cdot)\) \(\chi_{8712}(787,\cdot)\) \(\chi_{8712}(931,\cdot)\) \(\chi_{8712}(1003,\cdot)\) \(\chi_{8712}(1051,\cdot)\) \(\chi_{8712}(1075,\cdot)\) \(\chi_{8712}(1195,\cdot)\) \(\chi_{8712}(1267,\cdot)\) \(\chi_{8712}(1339,\cdot)\) \(\chi_{8712}(1579,\cdot)\) \(\chi_{8712}(1723,\cdot)\) \(\chi_{8712}(1795,\cdot)\) \(\chi_{8712}(1843,\cdot)\) \(\chi_{8712}(1867,\cdot)\) \(\chi_{8712}(1987,\cdot)\) \(\chi_{8712}(2059,\cdot)\) \(\chi_{8712}(2131,\cdot)\) \(\chi_{8712}(2371,\cdot)\) \(\chi_{8712}(2515,\cdot)\) \(\chi_{8712}(2587,\cdot)\) \(\chi_{8712}(2779,\cdot)\) \(\chi_{8712}(2851,\cdot)\) \(\chi_{8712}(2923,\cdot)\) \(\chi_{8712}(3163,\cdot)\) \(\chi_{8712}(3427,\cdot)\) \(\chi_{8712}(3451,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((6535,4357,1937,5689)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{67}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8712 }(139, a) \) \(1\)\(1\)\(e\left(\frac{79}{330}\right)\)\(e\left(\frac{16}{165}\right)\)\(e\left(\frac{113}{165}\right)\)\(e\left(\frac{93}{110}\right)\)\(e\left(\frac{61}{110}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{79}{165}\right)\)\(e\left(\frac{31}{165}\right)\)\(e\left(\frac{181}{330}\right)\)\(e\left(\frac{37}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8712 }(139,a) \;\) at \(\;a = \) e.g. 2