Properties

Label 640.613
Modulus $640$
Conductor $640$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

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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([0,9,24]))
 
pari: [g,chi] = znchar(Mod(613,640))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 640.bq

\(\chi_{640}(53,\cdot)\) \(\chi_{640}(77,\cdot)\) \(\chi_{640}(133,\cdot)\) \(\chi_{640}(157,\cdot)\) \(\chi_{640}(213,\cdot)\) \(\chi_{640}(237,\cdot)\) \(\chi_{640}(293,\cdot)\) \(\chi_{640}(317,\cdot)\) \(\chi_{640}(373,\cdot)\) \(\chi_{640}(397,\cdot)\) \(\chi_{640}(453,\cdot)\) \(\chi_{640}(477,\cdot)\) \(\chi_{640}(533,\cdot)\) \(\chi_{640}(557,\cdot)\) \(\chi_{640}(613,\cdot)\) \(\chi_{640}(637,\cdot)\)

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: 32.0.187072209578355573530071658587684226515959365500928000000000000000000000000.1

Values on generators

\((511,261,257)\) → \((1,e\left(\frac{9}{32}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 640 }(613, a) \) \(-1\)\(1\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{9}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 640 }(613,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 640 }(613,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 640 }(613,·),\chi_{ 640 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 640 }(613,·)) \;\) at \(\; a,b = \) e.g. 1,2