Properties

Label 127.66
Modulus $127$
Conductor $127$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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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([5]))
 
pari: [g,chi] = znchar(Mod(66,127))
 

Basic properties

Modulus: \(127\)
Conductor: \(127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 127.j

\(\chi_{127}(5,\cdot)\) \(\chi_{127}(10,\cdot)\) \(\chi_{127}(27,\cdot)\) \(\chi_{127}(33,\cdot)\) \(\chi_{127}(40,\cdot)\) \(\chi_{127}(51,\cdot)\) \(\chi_{127}(54,\cdot)\) \(\chi_{127}(66,\cdot)\) \(\chi_{127}(77,\cdot)\) \(\chi_{127}(80,\cdot)\) \(\chi_{127}(89,\cdot)\) \(\chi_{127}(102,\cdot)\)

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 42 polynomial

Values on generators

\(3\) → \(e\left(\frac{5}{42}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 127 }(66, a) \) \(-1\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{2}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 127 }(66,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 127 }(66,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 127 }(66,·),\chi_{ 127 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 127 }(66,·)) \;\) at \(\; a,b = \) e.g. 1,2