Properties

Label 4032.3835
Modulus $4032$
Conductor $448$
Order $16$
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(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,1,0,8]))
 
pari: [g,chi] = znchar(Mod(3835,4032))
 

Basic properties

Modulus: \(4032\)
Conductor: \(448\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{448}(251,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4032.fb

\(\chi_{4032}(307,\cdot)\) \(\chi_{4032}(811,\cdot)\) \(\chi_{4032}(1315,\cdot)\) \(\chi_{4032}(1819,\cdot)\) \(\chi_{4032}(2323,\cdot)\) \(\chi_{4032}(2827,\cdot)\) \(\chi_{4032}(3331,\cdot)\) \(\chi_{4032}(3835,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.16.3484608386920116940487669055488.4

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 4032 }(3835, a) \) \(1\)\(1\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(i\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(-1\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4032 }(3835,a) \;\) at \(\;a = \) e.g. 2