Properties

Label 179.3
Modulus $179$
Conductor $179$
Order $89$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(179, base_ring=CyclotomicField(178))
 
M = H._module
 
chi = DirichletCharacter(H, M([108]))
 
pari: [g,chi] = znchar(Mod(3,179))
 

Basic properties

Modulus: \(179\)
Conductor: \(179\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(89\)
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 179.c

\(\chi_{179}(3,\cdot)\) \(\chi_{179}(4,\cdot)\) \(\chi_{179}(5,\cdot)\) \(\chi_{179}(9,\cdot)\) \(\chi_{179}(12,\cdot)\) \(\chi_{179}(13,\cdot)\) \(\chi_{179}(14,\cdot)\) \(\chi_{179}(15,\cdot)\) \(\chi_{179}(16,\cdot)\) \(\chi_{179}(17,\cdot)\) \(\chi_{179}(19,\cdot)\) \(\chi_{179}(20,\cdot)\) \(\chi_{179}(22,\cdot)\) \(\chi_{179}(25,\cdot)\) \(\chi_{179}(27,\cdot)\) \(\chi_{179}(29,\cdot)\) \(\chi_{179}(31,\cdot)\) \(\chi_{179}(36,\cdot)\) \(\chi_{179}(39,\cdot)\) \(\chi_{179}(42,\cdot)\) \(\chi_{179}(43,\cdot)\) \(\chi_{179}(45,\cdot)\) \(\chi_{179}(46,\cdot)\) \(\chi_{179}(47,\cdot)\) \(\chi_{179}(48,\cdot)\) \(\chi_{179}(49,\cdot)\) \(\chi_{179}(51,\cdot)\) \(\chi_{179}(52,\cdot)\) \(\chi_{179}(56,\cdot)\) \(\chi_{179}(57,\cdot)\) ...

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: $\Q(\zeta_{89})$
Fixed field: Number field defined by a degree 89 polynomial

Values on generators

\(2\) → \(e\left(\frac{54}{89}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 179 }(3, a) \) \(1\)\(1\)\(e\left(\frac{54}{89}\right)\)\(e\left(\frac{47}{89}\right)\)\(e\left(\frac{19}{89}\right)\)\(e\left(\frac{65}{89}\right)\)\(e\left(\frac{12}{89}\right)\)\(e\left(\frac{67}{89}\right)\)\(e\left(\frac{73}{89}\right)\)\(e\left(\frac{5}{89}\right)\)\(e\left(\frac{30}{89}\right)\)\(e\left(\frac{9}{89}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 179 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 179 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 179 }(3,·),\chi_{ 179 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 179 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2