Properties

Label 4032.599
Modulus $4032$
Conductor $2016$
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([12,21,20,16]))
 
pari: [g,chi] = znchar(Mod(599,4032))
 

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 \(\chi_{2016}(851,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4032.fs

\(\chi_{4032}(599,\cdot)\) \(\chi_{4032}(695,\cdot)\) \(\chi_{4032}(1607,\cdot)\) \(\chi_{4032}(1703,\cdot)\) \(\chi_{4032}(2615,\cdot)\) \(\chi_{4032}(2711,\cdot)\) \(\chi_{4032}(3623,\cdot)\) \(\chi_{4032}(3719,\cdot)\)

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: Number field defined by a degree 24 polynomial

Values on generators

\((127,3781,1793,577)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(599, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{23}{24}\right)\)\(i\)\(-i\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(599,a) \;\) at \(\;a = \) e.g. 2