Properties

Label 533.149
Modulus $533$
Conductor $533$
Order $120$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(533, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,51]))
 
pari: [g,chi] = znchar(Mod(149,533))
 

Basic properties

Modulus: \(533\)
Conductor: \(533\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
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 533.cd

\(\chi_{533}(6,\cdot)\) \(\chi_{533}(7,\cdot)\) \(\chi_{533}(11,\cdot)\) \(\chi_{533}(15,\cdot)\) \(\chi_{533}(19,\cdot)\) \(\chi_{533}(58,\cdot)\) \(\chi_{533}(67,\cdot)\) \(\chi_{533}(89,\cdot)\) \(\chi_{533}(97,\cdot)\) \(\chi_{533}(106,\cdot)\) \(\chi_{533}(111,\cdot)\) \(\chi_{533}(145,\cdot)\) \(\chi_{533}(149,\cdot)\) \(\chi_{533}(158,\cdot)\) \(\chi_{533}(176,\cdot)\) \(\chi_{533}(188,\cdot)\) \(\chi_{533}(193,\cdot)\) \(\chi_{533}(227,\cdot)\) \(\chi_{533}(240,\cdot)\) \(\chi_{533}(258,\cdot)\) \(\chi_{533}(280,\cdot)\) \(\chi_{533}(358,\cdot)\) \(\chi_{533}(362,\cdot)\) \(\chi_{533}(397,\cdot)\) \(\chi_{533}(423,\cdot)\) \(\chi_{533}(440,\cdot)\) \(\chi_{533}(457,\cdot)\) \(\chi_{533}(462,\cdot)\) \(\chi_{533}(470,\cdot)\) \(\chi_{533}(479,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((288,170)\) → \((e\left(\frac{5}{12}\right),e\left(\frac{17}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 533 }(149, a) \) \(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{61}{120}\right)\)\(e\left(\frac{19}{120}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{23}{120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 533 }(149,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 533 }(149,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 533 }(149,·),\chi_{ 533 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 533 }(149,·)) \;\) at \(\; a,b = \) e.g. 1,2