Properties

Label 79.25
Modulus $79$
Conductor $79$
Order $39$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([46]))
 
pari: [g,chi] = znchar(Mod(25,79))
 

Basic properties

Modulus: \(79\)
Conductor: \(79\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(39\)
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.g

\(\chi_{79}(2,\cdot)\) \(\chi_{79}(4,\cdot)\) \(\chi_{79}(5,\cdot)\) \(\chi_{79}(9,\cdot)\) \(\chi_{79}(11,\cdot)\) \(\chi_{79}(13,\cdot)\) \(\chi_{79}(16,\cdot)\) \(\chi_{79}(19,\cdot)\) \(\chi_{79}(20,\cdot)\) \(\chi_{79}(25,\cdot)\) \(\chi_{79}(26,\cdot)\) \(\chi_{79}(31,\cdot)\) \(\chi_{79}(32,\cdot)\) \(\chi_{79}(36,\cdot)\) \(\chi_{79}(40,\cdot)\) \(\chi_{79}(42,\cdot)\) \(\chi_{79}(44,\cdot)\) \(\chi_{79}(45,\cdot)\) \(\chi_{79}(49,\cdot)\) \(\chi_{79}(50,\cdot)\) \(\chi_{79}(51,\cdot)\) \(\chi_{79}(72,\cdot)\) \(\chi_{79}(73,\cdot)\) \(\chi_{79}(76,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 39 polynomial

Values on generators

\(3\) → \(e\left(\frac{23}{39}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 79 }(25, a) \) \(1\)\(1\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{23}{39}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{37}{39}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{7}{39}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{4}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 79 }(25,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 79 }(25,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 79 }(25,·),\chi_{ 79 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 79 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2