Properties

Label 927.265
Modulus $927$
Conductor $927$
Order $51$
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([34,98]))
 
pari: [g,chi] = znchar(Mod(265,927))
 

Basic properties

Modulus: \(927\)
Conductor: \(927\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(51\)
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.z

\(\chi_{927}(7,\cdot)\) \(\chi_{927}(49,\cdot)\) \(\chi_{927}(52,\cdot)\) \(\chi_{927}(121,\cdot)\) \(\chi_{927}(223,\cdot)\) \(\chi_{927}(247,\cdot)\) \(\chi_{927}(265,\cdot)\) \(\chi_{927}(274,\cdot)\) \(\chi_{927}(304,\cdot)\) \(\chi_{927}(313,\cdot)\) \(\chi_{927}(328,\cdot)\) \(\chi_{927}(337,\cdot)\) \(\chi_{927}(364,\cdot)\) \(\chi_{927}(367,\cdot)\) \(\chi_{927}(448,\cdot)\) \(\chi_{927}(472,\cdot)\) \(\chi_{927}(475,\cdot)\) \(\chi_{927}(517,\cdot)\) \(\chi_{927}(544,\cdot)\) \(\chi_{927}(547,\cdot)\) \(\chi_{927}(565,\cdot)\) \(\chi_{927}(598,\cdot)\) \(\chi_{927}(634,\cdot)\) \(\chi_{927}(643,\cdot)\) \(\chi_{927}(700,\cdot)\) \(\chi_{927}(709,\cdot)\) \(\chi_{927}(715,\cdot)\) \(\chi_{927}(736,\cdot)\) \(\chi_{927}(754,\cdot)\) \(\chi_{927}(850,\cdot)\) ...

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 51 polynomial

Values on generators

\((722,829)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{49}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 927 }(265, a) \) \(1\)\(1\)\(e\left(\frac{31}{51}\right)\)\(e\left(\frac{11}{51}\right)\)\(e\left(\frac{32}{51}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{43}{51}\right)\)\(e\left(\frac{40}{51}\right)\)\(e\left(\frac{22}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(265,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 927 }(265,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(265,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 927 }(265,·)) \;\) at \(\; a,b = \) e.g. 1,2