Properties

Label 32.21
Modulus $32$
Conductor $32$
Order $8$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5]))
 
pari: [g,chi] = znchar(Mod(21,32))
 

Basic properties

Modulus: \(32\)
Conductor: \(32\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 32.g

\(\chi_{32}(5,\cdot)\) \(\chi_{32}(13,\cdot)\) \(\chi_{32}(21,\cdot)\) \(\chi_{32}(29,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: \(\Q(\zeta_{32})^+\)

Values on generators

\((31,5)\) → \((1,e\left(\frac{5}{8}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 32 }(21, a) \) \(1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 32 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 32 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 32 }(21,·),\chi_{ 32 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 32 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2