Properties

Label 79.64
Modulus $79$
Conductor $79$
Order $13$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, base_ring=CyclotomicField(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(64,79))
 

Basic properties

Modulus: \(79\)
Conductor: \(79\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(13\)
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 79.e

\(\chi_{79}(8,\cdot)\) \(\chi_{79}(10,\cdot)\) \(\chi_{79}(18,\cdot)\) \(\chi_{79}(21,\cdot)\) \(\chi_{79}(22,\cdot)\) \(\chi_{79}(38,\cdot)\) \(\chi_{79}(46,\cdot)\) \(\chi_{79}(52,\cdot)\) \(\chi_{79}(62,\cdot)\) \(\chi_{79}(64,\cdot)\) \(\chi_{79}(65,\cdot)\) \(\chi_{79}(67,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 13 polynomial

Values on generators

\(3\) → \(e\left(\frac{4}{13}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 79 }(64, a) \) \(1\)\(1\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{12}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 79 }(64,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 79 }(64,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 79 }(64,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 79 }(64,·)) \;\) at \(\; a,b = \) e.g. 1,2