Properties

Label 139.22
Modulus $139$
Conductor $139$
Order $138$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(139, base_ring=CyclotomicField(138))
 
M = H._module
 
chi = DirichletCharacter(H, M([77]))
 
pari: [g,chi] = znchar(Mod(22,139))
 

Basic properties

Modulus: \(139\)
Conductor: \(139\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(138\)
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 139.h

\(\chi_{139}(2,\cdot)\) \(\chi_{139}(3,\cdot)\) \(\chi_{139}(12,\cdot)\) \(\chi_{139}(15,\cdot)\) \(\chi_{139}(17,\cdot)\) \(\chi_{139}(18,\cdot)\) \(\chi_{139}(19,\cdot)\) \(\chi_{139}(21,\cdot)\) \(\chi_{139}(22,\cdot)\) \(\chi_{139}(26,\cdot)\) \(\chi_{139}(32,\cdot)\) \(\chi_{139}(40,\cdot)\) \(\chi_{139}(50,\cdot)\) \(\chi_{139}(53,\cdot)\) \(\chi_{139}(56,\cdot)\) \(\chi_{139}(58,\cdot)\) \(\chi_{139}(61,\cdot)\) \(\chi_{139}(68,\cdot)\) \(\chi_{139}(70,\cdot)\) \(\chi_{139}(72,\cdot)\) \(\chi_{139}(73,\cdot)\) \(\chi_{139}(85,\cdot)\) \(\chi_{139}(88,\cdot)\) \(\chi_{139}(90,\cdot)\) \(\chi_{139}(92,\cdot)\) \(\chi_{139}(93,\cdot)\) \(\chi_{139}(98,\cdot)\) \(\chi_{139}(101,\cdot)\) \(\chi_{139}(102,\cdot)\) \(\chi_{139}(104,\cdot)\) ...

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: $\Q(\zeta_{69})$
Fixed field: Number field defined by a degree 138 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{77}{138}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 139 }(22, a) \) \(-1\)\(1\)\(e\left(\frac{77}{138}\right)\)\(e\left(\frac{121}{138}\right)\)\(e\left(\frac{8}{69}\right)\)\(e\left(\frac{68}{69}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{62}{69}\right)\)\(e\left(\frac{31}{46}\right)\)\(e\left(\frac{52}{69}\right)\)\(e\left(\frac{25}{46}\right)\)\(e\left(\frac{28}{69}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 139 }(22,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 139 }(22,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 139 }(22,·),\chi_{ 139 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 139 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2