Properties

Label 4032.3971
Modulus $4032$
Conductor $4032$
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(4032, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,9,8,16]))
 
pari: [g,chi] = znchar(Mod(3971,4032))
 

Basic properties

Modulus: \(4032\)
Conductor: \(4032\)
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 4032.ha

\(\chi_{4032}(347,\cdot)\) \(\chi_{4032}(443,\cdot)\) \(\chi_{4032}(851,\cdot)\) \(\chi_{4032}(947,\cdot)\) \(\chi_{4032}(1355,\cdot)\) \(\chi_{4032}(1451,\cdot)\) \(\chi_{4032}(1859,\cdot)\) \(\chi_{4032}(1955,\cdot)\) \(\chi_{4032}(2363,\cdot)\) \(\chi_{4032}(2459,\cdot)\) \(\chi_{4032}(2867,\cdot)\) \(\chi_{4032}(2963,\cdot)\) \(\chi_{4032}(3371,\cdot)\) \(\chi_{4032}(3467,\cdot)\) \(\chi_{4032}(3875,\cdot)\) \(\chi_{4032}(3971,\cdot)\)

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

\((127,3781,1793,577)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(3971, a) \) \(1\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(3971,a) \;\) at \(\;a = \) e.g. 2