Properties

Label 9072.139
Modulus $9072$
Conductor $9072$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9072, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([54,27,76,54]))
 
pari: [g,chi] = znchar(Mod(139,9072))
 

Basic properties

Modulus: \(9072\)
Conductor: \(9072\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
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 9072.jz

\(\chi_{9072}(139,\cdot)\) \(\chi_{9072}(475,\cdot)\) \(\chi_{9072}(643,\cdot)\) \(\chi_{9072}(979,\cdot)\) \(\chi_{9072}(1147,\cdot)\) \(\chi_{9072}(1483,\cdot)\) \(\chi_{9072}(1651,\cdot)\) \(\chi_{9072}(1987,\cdot)\) \(\chi_{9072}(2155,\cdot)\) \(\chi_{9072}(2491,\cdot)\) \(\chi_{9072}(2659,\cdot)\) \(\chi_{9072}(2995,\cdot)\) \(\chi_{9072}(3163,\cdot)\) \(\chi_{9072}(3499,\cdot)\) \(\chi_{9072}(3667,\cdot)\) \(\chi_{9072}(4003,\cdot)\) \(\chi_{9072}(4171,\cdot)\) \(\chi_{9072}(4507,\cdot)\) \(\chi_{9072}(4675,\cdot)\) \(\chi_{9072}(5011,\cdot)\) \(\chi_{9072}(5179,\cdot)\) \(\chi_{9072}(5515,\cdot)\) \(\chi_{9072}(5683,\cdot)\) \(\chi_{9072}(6019,\cdot)\) \(\chi_{9072}(6187,\cdot)\) \(\chi_{9072}(6523,\cdot)\) \(\chi_{9072}(6691,\cdot)\) \(\chi_{9072}(7027,\cdot)\) \(\chi_{9072}(7195,\cdot)\) \(\chi_{9072}(7531,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((1135,6805,3809,2593)\) → \((-1,i,e\left(\frac{19}{27}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 9072 }(139, a) \) \(1\)\(1\)\(e\left(\frac{101}{108}\right)\)\(e\left(\frac{97}{108}\right)\)\(e\left(\frac{95}{108}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{47}{54}\right)\)\(e\left(\frac{85}{108}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{29}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9072 }(139,a) \;\) at \(\;a = \) e.g. 2