Properties

Label 151.58
Modulus $151$
Conductor $151$
Order $75$
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(150))
 
M = H._module
 
chi = DirichletCharacter(H, M([34]))
 
pari: [g,chi] = znchar(Mod(58,151))
 

Basic properties

Modulus: \(151\)
Conductor: \(151\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(75\)
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.k

\(\chi_{151}(5,\cdot)\) \(\chi_{151}(10,\cdot)\) \(\chi_{151}(11,\cdot)\) \(\chi_{151}(17,\cdot)\) \(\chi_{151}(18,\cdot)\) \(\chi_{151}(21,\cdot)\) \(\chi_{151}(22,\cdot)\) \(\chi_{151}(25,\cdot)\) \(\chi_{151}(31,\cdot)\) \(\chi_{151}(34,\cdot)\) \(\chi_{151}(36,\cdot)\) \(\chi_{151}(37,\cdot)\) \(\chi_{151}(39,\cdot)\) \(\chi_{151}(40,\cdot)\) \(\chi_{151}(42,\cdot)\) \(\chi_{151}(43,\cdot)\) \(\chi_{151}(45,\cdot)\) \(\chi_{151}(47,\cdot)\) \(\chi_{151}(49,\cdot)\) \(\chi_{151}(55,\cdot)\) \(\chi_{151}(58,\cdot)\) \(\chi_{151}(62,\cdot)\) \(\chi_{151}(69,\cdot)\) \(\chi_{151}(74,\cdot)\) \(\chi_{151}(80,\cdot)\) \(\chi_{151}(88,\cdot)\) \(\chi_{151}(90,\cdot)\) \(\chi_{151}(95,\cdot)\) \(\chi_{151}(97,\cdot)\) \(\chi_{151}(99,\cdot)\) ...

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: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 75 polynomial

Values on generators

\(6\) → \(e\left(\frac{17}{75}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 151 }(58, a) \) \(1\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{29}{75}\right)\)\(e\left(\frac{17}{75}\right)\)\(e\left(\frac{14}{75}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{19}{75}\right)\)\(e\left(\frac{23}{75}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 151 }(58,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 151 }(58,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 151 }(58,·),\chi_{ 151 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 151 }(58,·)) \;\) at \(\; a,b = \) e.g. 1,2