Properties

Label 4032.3767
Modulus $4032$
Conductor $288$
Order $24$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,9,20,0]))
 
pari: [g,chi] = znchar(Mod(3767,4032))
 

Basic properties

Modulus: \(4032\)
Conductor: \(288\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{288}(131,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4032.gl

\(\chi_{4032}(407,\cdot)\) \(\chi_{4032}(743,\cdot)\) \(\chi_{4032}(1415,\cdot)\) \(\chi_{4032}(1751,\cdot)\) \(\chi_{4032}(2423,\cdot)\) \(\chi_{4032}(2759,\cdot)\) \(\chi_{4032}(3431,\cdot)\) \(\chi_{4032}(3767,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.24.1486465269728735333725176976133731985582456832.1

Values on generators

\((127,3781,1793,577)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{5}{6}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(3767, a) \) \(1\)\(1\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(3767,a) \;\) at \(\;a = \) e.g. 2