Properties

Label 24704.9
Modulus $24704$
Conductor $12352$
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(24704, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,14]))
 
pari: [g,chi] = znchar(Mod(9,24704))
 

Basic properties

Modulus: \(24704\)
Conductor: \(12352\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{12352}(10045,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 24704.dr

\(\chi_{24704}(9,\cdot)\) \(\chi_{24704}(729,\cdot)\) \(\chi_{24704}(2745,\cdot)\) \(\chi_{24704}(9641,\cdot)\) \(\chi_{24704}(12361,\cdot)\) \(\chi_{24704}(13081,\cdot)\) \(\chi_{24704}(15097,\cdot)\) \(\chi_{24704}(21993,\cdot)\)

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: Number field defined by a degree 16 polynomial

Values on generators

\((24319,773,23937)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 24704 }(9, a) \) \(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{15}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 24704 }(9,a) \;\) at \(\;a = \) e.g. 2