Properties

Label 9216.575
Modulus $9216$
Conductor $192$
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(9216, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,9,8]))
 
pari: [g,chi] = znchar(Mod(575,9216))
 

Basic properties

Modulus: \(9216\)
Conductor: \(192\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{192}(155,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9216.be

\(\chi_{9216}(575,\cdot)\) \(\chi_{9216}(1727,\cdot)\) \(\chi_{9216}(2879,\cdot)\) \(\chi_{9216}(4031,\cdot)\) \(\chi_{9216}(5183,\cdot)\) \(\chi_{9216}(6335,\cdot)\) \(\chi_{9216}(7487,\cdot)\) \(\chi_{9216}(8639,\cdot)\)

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: 16.16.3965881151245791007623610368.1

Values on generators

\((8191,2053,4097)\) → \((-1,e\left(\frac{9}{16}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 9216 }(575, a) \) \(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(i\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9216 }(575,a) \;\) at \(\;a = \) e.g. 2