Properties

Label 640.107
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([16,13,8]))
 
pari: [g,chi] = znchar(Mod(107,640))
 

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.bl

\(\chi_{640}(3,\cdot)\) \(\chi_{640}(27,\cdot)\) \(\chi_{640}(83,\cdot)\) \(\chi_{640}(107,\cdot)\) \(\chi_{640}(163,\cdot)\) \(\chi_{640}(187,\cdot)\) \(\chi_{640}(243,\cdot)\) \(\chi_{640}(267,\cdot)\) \(\chi_{640}(323,\cdot)\) \(\chi_{640}(347,\cdot)\) \(\chi_{640}(403,\cdot)\) \(\chi_{640}(427,\cdot)\) \(\chi_{640}(483,\cdot)\) \(\chi_{640}(507,\cdot)\) \(\chi_{640}(563,\cdot)\) \(\chi_{640}(587,\cdot)\)

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: 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.2

Values on generators

\((511,261,257)\) → \((-1,e\left(\frac{13}{32}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 640 }(107, a) \) \(1\)\(1\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{13}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 640 }(107,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 640 }(107,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 640 }(107,·),\chi_{ 640 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 640 }(107,·)) \;\) at \(\; a,b = \) e.g. 1,2