Properties

Label 181.178
Modulus $181$
Conductor $181$
Order $90$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(181, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([73]))
 
pari: [g,chi] = znchar(Mod(178,181))
 

Basic properties

Modulus: \(181\)
Conductor: \(181\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 181.q

\(\chi_{181}(4,\cdot)\) \(\chi_{181}(11,\cdot)\) \(\chi_{181}(12,\cdot)\) \(\chi_{181}(20,\cdot)\) \(\chi_{181}(33,\cdot)\) \(\chi_{181}(37,\cdot)\) \(\chi_{181}(52,\cdot)\) \(\chi_{181}(55,\cdot)\) \(\chi_{181}(60,\cdot)\) \(\chi_{181}(79,\cdot)\) \(\chi_{181}(94,\cdot)\) \(\chi_{181}(100,\cdot)\) \(\chi_{181}(106,\cdot)\) \(\chi_{181}(111,\cdot)\) \(\chi_{181}(136,\cdot)\) \(\chi_{181}(137,\cdot)\) \(\chi_{181}(143,\cdot)\) \(\chi_{181}(147,\cdot)\) \(\chi_{181}(165,\cdot)\) \(\chi_{181}(166,\cdot)\) \(\chi_{181}(167,\cdot)\) \(\chi_{181}(168,\cdot)\) \(\chi_{181}(172,\cdot)\) \(\chi_{181}(178,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\(2\) → \(e\left(\frac{73}{90}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 181 }(178, a) \) \(1\)\(1\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{38}{45}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{13}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 181 }(178,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 181 }(178,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 181 }(178,·),\chi_{ 181 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 181 }(178,·)) \;\) at \(\; a,b = \) e.g. 1,2