Properties

Label 9792.403
Modulus $9792$
Conductor $9792$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,21,32,39]))
 
pari: [g,chi] = znchar(Mod(403,9792))
 

Basic properties

Modulus: \(9792\)
Conductor: \(9792\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 9792.ob

\(\chi_{9792}(403,\cdot)\) \(\chi_{9792}(2203,\cdot)\) \(\chi_{9792}(2275,\cdot)\) \(\chi_{9792}(2419,\cdot)\) \(\chi_{9792}(3067,\cdot)\) \(\chi_{9792}(3667,\cdot)\) \(\chi_{9792}(4363,\cdot)\) \(\chi_{9792}(5443,\cdot)\) \(\chi_{9792}(5467,\cdot)\) \(\chi_{9792}(5539,\cdot)\) \(\chi_{9792}(5683,\cdot)\) \(\chi_{9792}(5803,\cdot)\) \(\chi_{9792}(6331,\cdot)\) \(\chi_{9792}(7627,\cdot)\) \(\chi_{9792}(8707,\cdot)\) \(\chi_{9792}(9067,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{13}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 9792 }(403, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{5}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9792 }(403,a) \;\) at \(\;a = \) e.g. 2