Properties

Label 164.47
Modulus $164$
Conductor $164$
Order $40$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,1]))
 
pari: [g,chi] = znchar(Mod(47,164))
 

Basic properties

Modulus: \(164\)
Conductor: \(164\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
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 164.o

\(\chi_{164}(7,\cdot)\) \(\chi_{164}(11,\cdot)\) \(\chi_{164}(15,\cdot)\) \(\chi_{164}(19,\cdot)\) \(\chi_{164}(35,\cdot)\) \(\chi_{164}(47,\cdot)\) \(\chi_{164}(63,\cdot)\) \(\chi_{164}(67,\cdot)\) \(\chi_{164}(71,\cdot)\) \(\chi_{164}(75,\cdot)\) \(\chi_{164}(95,\cdot)\) \(\chi_{164}(99,\cdot)\) \(\chi_{164}(111,\cdot)\) \(\chi_{164}(135,\cdot)\) \(\chi_{164}(147,\cdot)\) \(\chi_{164}(151,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: \(\Q(\zeta_{164})^+\)

Values on generators

\((83,129)\) → \((-1,e\left(\frac{1}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 164 }(47, a) \) \(1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{19}{40}\right)\)\(-i\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{7}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 164 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 164 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 164 }(47,·),\chi_{ 164 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 164 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2