Properties

Label 805.73
Modulus $805$
Conductor $805$
Order $132$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([99,22,24]))
 
pari: [g,chi] = znchar(Mod(73,805))
 

Basic properties

Modulus: \(805\)
Conductor: \(805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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 805.bv

\(\chi_{805}(3,\cdot)\) \(\chi_{805}(12,\cdot)\) \(\chi_{805}(52,\cdot)\) \(\chi_{805}(73,\cdot)\) \(\chi_{805}(82,\cdot)\) \(\chi_{805}(87,\cdot)\) \(\chi_{805}(108,\cdot)\) \(\chi_{805}(117,\cdot)\) \(\chi_{805}(173,\cdot)\) \(\chi_{805}(187,\cdot)\) \(\chi_{805}(192,\cdot)\) \(\chi_{805}(213,\cdot)\) \(\chi_{805}(243,\cdot)\) \(\chi_{805}(248,\cdot)\) \(\chi_{805}(257,\cdot)\) \(\chi_{805}(262,\cdot)\) \(\chi_{805}(278,\cdot)\) \(\chi_{805}(292,\cdot)\) \(\chi_{805}(348,\cdot)\) \(\chi_{805}(353,\cdot)\) \(\chi_{805}(397,\cdot)\) \(\chi_{805}(418,\cdot)\) \(\chi_{805}(423,\cdot)\) \(\chi_{805}(432,\cdot)\) \(\chi_{805}(453,\cdot)\) \(\chi_{805}(472,\cdot)\) \(\chi_{805}(537,\cdot)\) \(\chi_{805}(542,\cdot)\) \(\chi_{805}(558,\cdot)\) \(\chi_{805}(577,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((162,346,281)\) → \((-i,e\left(\frac{1}{6}\right),e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 805 }(73, a) \) \(1\)\(1\)\(e\left(\frac{59}{132}\right)\)\(e\left(\frac{43}{132}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{29}{132}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{26}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 805 }(73,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 805 }(73,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 805 }(73,·),\chi_{ 805 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 805 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2