Properties

Label 1088.79
Modulus $1088$
Conductor $272$
Order $16$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,4,7]))
 
pari: [g,chi] = znchar(Mod(79,1088))
 

Basic properties

Modulus: \(1088\)
Conductor: \(272\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{272}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1088.cd

\(\chi_{1088}(79,\cdot)\) \(\chi_{1088}(143,\cdot)\) \(\chi_{1088}(175,\cdot)\) \(\chi_{1088}(303,\cdot)\) \(\chi_{1088}(751,\cdot)\) \(\chi_{1088}(879,\cdot)\) \(\chi_{1088}(911,\cdot)\) \(\chi_{1088}(975,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.16.50356278859985642512053476261888.2

Values on generators

\((511,69,513)\) → \((-1,i,e\left(\frac{7}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 1088 }(79, a) \) \(1\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1088 }(79,a) \;\) at \(\;a = \) e.g. 2