Properties

Label 4032.3707
Modulus $4032$
Conductor $1344$
Order $48$
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(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,3,24,32]))
 
pari: [g,chi] = znchar(Mod(3707,4032))
 

Basic properties

Modulus: \(4032\)
Conductor: \(1344\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1344}(1019,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4032.gz

\(\chi_{4032}(107,\cdot)\) \(\chi_{4032}(179,\cdot)\) \(\chi_{4032}(611,\cdot)\) \(\chi_{4032}(683,\cdot)\) \(\chi_{4032}(1115,\cdot)\) \(\chi_{4032}(1187,\cdot)\) \(\chi_{4032}(1619,\cdot)\) \(\chi_{4032}(1691,\cdot)\) \(\chi_{4032}(2123,\cdot)\) \(\chi_{4032}(2195,\cdot)\) \(\chi_{4032}(2627,\cdot)\) \(\chi_{4032}(2699,\cdot)\) \(\chi_{4032}(3131,\cdot)\) \(\chi_{4032}(3203,\cdot)\) \(\chi_{4032}(3635,\cdot)\) \(\chi_{4032}(3707,\cdot)\)

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

\((127,3781,1793,577)\) → \((-1,e\left(\frac{1}{16}\right),-1,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(3707, a) \) \(1\)\(1\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{43}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(3707,a) \;\) at \(\;a = \) e.g. 2