Properties

Label 9128.8971
Modulus $9128$
Conductor $9128$
Order $54$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9128, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([27,27,36,34]))
 
pari: [g,chi] = znchar(Mod(8971,9128))
 

Basic properties

Modulus: \(9128\)
Conductor: \(9128\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9128.fg

\(\chi_{9128}(347,\cdot)\) \(\chi_{9128}(387,\cdot)\) \(\chi_{9128}(403,\cdot)\) \(\chi_{9128}(947,\cdot)\) \(\chi_{9128}(2307,\cdot)\) \(\chi_{9128}(2571,\cdot)\) \(\chi_{9128}(2907,\cdot)\) \(\chi_{9128}(3651,\cdot)\) \(\chi_{9128}(4027,\cdot)\) \(\chi_{9128}(4139,\cdot)\) \(\chi_{9128}(4547,\cdot)\) \(\chi_{9128}(5203,\cdot)\) \(\chi_{9128}(6003,\cdot)\) \(\chi_{9128}(6675,\cdot)\) \(\chi_{9128}(7683,\cdot)\) \(\chi_{9128}(8675,\cdot)\) \(\chi_{9128}(8971,\cdot)\) \(\chi_{9128}(9123,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((6847,4565,1305,7337)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{17}{27}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 9128 }(8971, a) \) \(-1\)\(1\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{27}\right)\)\(-1\)\(e\left(\frac{5}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9128 }(8971,a) \;\) at \(\;a = \) e.g. 2