Properties

Label 483.89
Modulus $483$
Conductor $483$
Order $66$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,55,15]))
 
pari: [g,chi] = znchar(Mod(89,483))
 

Basic properties

Modulus: \(483\)
Conductor: \(483\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 483.ba

\(\chi_{483}(5,\cdot)\) \(\chi_{483}(17,\cdot)\) \(\chi_{483}(38,\cdot)\) \(\chi_{483}(80,\cdot)\) \(\chi_{483}(89,\cdot)\) \(\chi_{483}(122,\cdot)\) \(\chi_{483}(143,\cdot)\) \(\chi_{483}(152,\cdot)\) \(\chi_{483}(194,\cdot)\) \(\chi_{483}(227,\cdot)\) \(\chi_{483}(290,\cdot)\) \(\chi_{483}(320,\cdot)\) \(\chi_{483}(332,\cdot)\) \(\chi_{483}(341,\cdot)\) \(\chi_{483}(362,\cdot)\) \(\chi_{483}(383,\cdot)\) \(\chi_{483}(425,\cdot)\) \(\chi_{483}(458,\cdot)\) \(\chi_{483}(467,\cdot)\) \(\chi_{483}(479,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((323,346,442)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 483 }(89, a) \) \(-1\)\(1\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{19}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 483 }(89,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 483 }(89,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 483 }(89,·),\chi_{ 483 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 483 }(89,·)) \;\) at \(\; a,b = \) e.g. 1,2