Properties

Label 127.38
Modulus $127$
Conductor $127$
Order $21$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(127, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([10]))
 
pari: [g,chi] = znchar(Mod(38,127))
 

Basic properties

Modulus: \(127\)
Conductor: \(127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
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 127.i

\(\chi_{127}(25,\cdot)\) \(\chi_{127}(38,\cdot)\) \(\chi_{127}(47,\cdot)\) \(\chi_{127}(50,\cdot)\) \(\chi_{127}(61,\cdot)\) \(\chi_{127}(73,\cdot)\) \(\chi_{127}(76,\cdot)\) \(\chi_{127}(87,\cdot)\) \(\chi_{127}(94,\cdot)\) \(\chi_{127}(100,\cdot)\) \(\chi_{127}(117,\cdot)\) \(\chi_{127}(122,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\(3\) → \(e\left(\frac{5}{21}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 127 }(38, a) \) \(1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{4}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 127 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 127 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 127 }(38,·),\chi_{ 127 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 127 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2