Properties

Label 233.19
Modulus $233$
Conductor $233$
Order $29$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(233, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([34]))
 
pari: [g,chi] = znchar(Mod(19,233))
 

Basic properties

Modulus: \(233\)
Conductor: \(233\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(29\)
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 233.e

\(\chi_{233}(2,\cdot)\) \(\chi_{233}(4,\cdot)\) \(\chi_{233}(8,\cdot)\) \(\chi_{233}(16,\cdot)\) \(\chi_{233}(19,\cdot)\) \(\chi_{233}(23,\cdot)\) \(\chi_{233}(32,\cdot)\) \(\chi_{233}(37,\cdot)\) \(\chi_{233}(38,\cdot)\) \(\chi_{233}(46,\cdot)\) \(\chi_{233}(51,\cdot)\) \(\chi_{233}(63,\cdot)\) \(\chi_{233}(64,\cdot)\) \(\chi_{233}(71,\cdot)\) \(\chi_{233}(74,\cdot)\) \(\chi_{233}(76,\cdot)\) \(\chi_{233}(92,\cdot)\) \(\chi_{233}(102,\cdot)\) \(\chi_{233}(117,\cdot)\) \(\chi_{233}(126,\cdot)\) \(\chi_{233}(128,\cdot)\) \(\chi_{233}(135,\cdot)\) \(\chi_{233}(142,\cdot)\) \(\chi_{233}(148,\cdot)\) \(\chi_{233}(152,\cdot)\) \(\chi_{233}(175,\cdot)\) \(\chi_{233}(184,\cdot)\) \(\chi_{233}(204,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 29 polynomial

Values on generators

\(3\) → \(e\left(\frac{17}{29}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 233 }(19, a) \) \(1\)\(1\)\(e\left(\frac{6}{29}\right)\)\(e\left(\frac{17}{29}\right)\)\(e\left(\frac{12}{29}\right)\)\(e\left(\frac{21}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{4}{29}\right)\)\(e\left(\frac{18}{29}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{14}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 233 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 233 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 233 }(19,·),\chi_{ 233 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 233 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2