Properties

Label 4032.4019
Modulus $4032$
Conductor $576$
Order $48$
Real no
Primitive no
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([24,45,40,0]))
 
pari: [g,chi] = znchar(Mod(4019,4032))
 

Basic properties

Modulus: \(4032\)
Conductor: \(576\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{576}(563,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4032.hb

\(\chi_{4032}(155,\cdot)\) \(\chi_{4032}(491,\cdot)\) \(\chi_{4032}(659,\cdot)\) \(\chi_{4032}(995,\cdot)\) \(\chi_{4032}(1163,\cdot)\) \(\chi_{4032}(1499,\cdot)\) \(\chi_{4032}(1667,\cdot)\) \(\chi_{4032}(2003,\cdot)\) \(\chi_{4032}(2171,\cdot)\) \(\chi_{4032}(2507,\cdot)\) \(\chi_{4032}(2675,\cdot)\) \(\chi_{4032}(3011,\cdot)\) \(\chi_{4032}(3179,\cdot)\) \(\chi_{4032}(3515,\cdot)\) \(\chi_{4032}(3683,\cdot)\) \(\chi_{4032}(4019,\cdot)\)

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

\((127,3781,1793,577)\) → \((-1,e\left(\frac{15}{16}\right),e\left(\frac{5}{6}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(4019, a) \) \(1\)\(1\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{35}{48}\right)\)\(-i\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(4019,a) \;\) at \(\;a = \) e.g. 2