Properties

Label 151.4
Modulus $151$
Conductor $151$
Order $15$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(151, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([28]))
 
pari: [g,chi] = znchar(Mod(4,151))
 

Basic properties

Modulus: \(151\)
Conductor: \(151\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
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 151.g

\(\chi_{151}(2,\cdot)\) \(\chi_{151}(4,\cdot)\) \(\chi_{151}(16,\cdot)\) \(\chi_{151}(38,\cdot)\) \(\chi_{151}(76,\cdot)\) \(\chi_{151}(85,\cdot)\) \(\chi_{151}(105,\cdot)\) \(\chi_{151}(128,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\(6\) → \(e\left(\frac{14}{15}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 151 }(4, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 151 }(4,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 151 }(4,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 151 }(4,·),\chi_{ 151 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 151 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2