Properties

Label 4032.2729
Modulus $4032$
Conductor $2016$
Order $24$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,4,12]))
 
pari: [g,chi] = znchar(Mod(2729,4032))
 

Basic properties

Modulus: \(4032\)
Conductor: \(2016\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2016}(461,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4032.fj

\(\chi_{4032}(41,\cdot)\) \(\chi_{4032}(713,\cdot)\) \(\chi_{4032}(1049,\cdot)\) \(\chi_{4032}(1721,\cdot)\) \(\chi_{4032}(2057,\cdot)\) \(\chi_{4032}(2729,\cdot)\) \(\chi_{4032}(3065,\cdot)\) \(\chi_{4032}(3737,\cdot)\)

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: 24.24.20574592712627357116606755731219706996406776278896607232.1

Values on generators

\((127,3781,1793,577)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{1}{6}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(2729, a) \) \(1\)\(1\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(-1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(2729,a) \;\) at \(\;a = \) e.g. 2