Properties

Label 9792.449
Modulus $9792$
Conductor $51$
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(9792, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,8,11]))
 
pari: [g,chi] = znchar(Mod(449,9792))
 

Basic properties

Modulus: \(9792\)
Conductor: \(51\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{51}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9792.gn

\(\chi_{9792}(449,\cdot)\) \(\chi_{9792}(1025,\cdot)\) \(\chi_{9792}(1601,\cdot)\) \(\chi_{9792}(3329,\cdot)\) \(\chi_{9792}(3905,\cdot)\) \(\chi_{9792}(4481,\cdot)\) \(\chi_{9792}(5633,\cdot)\) \(\chi_{9792}(9089,\cdot)\)

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: \(\Q(\zeta_{51})^+\)

Values on generators

\((7039,5509,8705,9217)\) → \((1,1,-1,e\left(\frac{11}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 9792 }(449, a) \) \(1\)\(1\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9792 }(449,a) \;\) at \(\;a = \) e.g. 2