Properties

Label 640.523
Modulus $640$
Conductor $640$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,21,24]))
 
pari: [g,chi] = znchar(Mod(523,640))
 

Basic properties

Modulus: \(640\)
Conductor: \(640\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 640.br

\(\chi_{640}(43,\cdot)\) \(\chi_{640}(67,\cdot)\) \(\chi_{640}(123,\cdot)\) \(\chi_{640}(147,\cdot)\) \(\chi_{640}(203,\cdot)\) \(\chi_{640}(227,\cdot)\) \(\chi_{640}(283,\cdot)\) \(\chi_{640}(307,\cdot)\) \(\chi_{640}(363,\cdot)\) \(\chi_{640}(387,\cdot)\) \(\chi_{640}(443,\cdot)\) \(\chi_{640}(467,\cdot)\) \(\chi_{640}(523,\cdot)\) \(\chi_{640}(547,\cdot)\) \(\chi_{640}(603,\cdot)\) \(\chi_{640}(627,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.1

Values on generators

\((511,261,257)\) → \((-1,e\left(\frac{21}{32}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 640 }(523, a) \) \(1\)\(1\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 640 }(523,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 640 }(523,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 640 }(523,·),\chi_{ 640 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 640 }(523,·)) \;\) at \(\; a,b = \) e.g. 1,2