Properties

Label 927.11
Modulus $927$
Conductor $927$
Order $102$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([17,61]))
 
pari: [g,chi] = znchar(Mod(11,927))
 

Basic properties

Modulus: \(927\)
Conductor: \(927\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
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 927.bg

\(\chi_{927}(11,\cdot)\) \(\chi_{927}(65,\cdot)\) \(\chi_{927}(77,\cdot)\) \(\chi_{927}(173,\cdot)\) \(\chi_{927}(191,\cdot)\) \(\chi_{927}(212,\cdot)\) \(\chi_{927}(218,\cdot)\) \(\chi_{927}(227,\cdot)\) \(\chi_{927}(284,\cdot)\) \(\chi_{927}(293,\cdot)\) \(\chi_{927}(329,\cdot)\) \(\chi_{927}(362,\cdot)\) \(\chi_{927}(380,\cdot)\) \(\chi_{927}(383,\cdot)\) \(\chi_{927}(410,\cdot)\) \(\chi_{927}(452,\cdot)\) \(\chi_{927}(455,\cdot)\) \(\chi_{927}(479,\cdot)\) \(\chi_{927}(560,\cdot)\) \(\chi_{927}(563,\cdot)\) \(\chi_{927}(590,\cdot)\) \(\chi_{927}(599,\cdot)\) \(\chi_{927}(614,\cdot)\) \(\chi_{927}(623,\cdot)\) \(\chi_{927}(653,\cdot)\) \(\chi_{927}(662,\cdot)\) \(\chi_{927}(680,\cdot)\) \(\chi_{927}(704,\cdot)\) \(\chi_{927}(806,\cdot)\) \(\chi_{927}(875,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((722,829)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{61}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 927 }(11, a) \) \(1\)\(1\)\(e\left(\frac{49}{102}\right)\)\(e\left(\frac{49}{51}\right)\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{31}{34}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{20}{51}\right)\)\(e\left(\frac{55}{102}\right)\)\(e\left(\frac{47}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 927 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(11,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 927 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2