Properties

Label 927.106
Modulus $927$
Conductor $927$
Order $102$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(927, base_ring=CyclotomicField(102))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([68,39]))
 
pari: [g,chi] = znchar(Mod(106,927))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 927.bl

\(\chi_{927}(22,\cdot)\) \(\chi_{927}(31,\cdot)\) \(\chi_{927}(94,\cdot)\) \(\chi_{927}(106,\cdot)\) \(\chi_{927}(130,\cdot)\) \(\chi_{927}(142,\cdot)\) \(\chi_{927}(193,\cdot)\) \(\chi_{927}(286,\cdot)\) \(\chi_{927}(295,\cdot)\) \(\chi_{927}(301,\cdot)\) \(\chi_{927}(319,\cdot)\) \(\chi_{927}(331,\cdot)\) \(\chi_{927}(340,\cdot)\) \(\chi_{927}(346,\cdot)\) \(\chi_{927}(382,\cdot)\) \(\chi_{927}(403,\cdot)\) \(\chi_{927}(436,\cdot)\) \(\chi_{927}(439,\cdot)\) \(\chi_{927}(454,\cdot)\) \(\chi_{927}(481,\cdot)\) \(\chi_{927}(502,\cdot)\) \(\chi_{927}(610,\cdot)\) \(\chi_{927}(628,\cdot)\) \(\chi_{927}(655,\cdot)\) \(\chi_{927}(691,\cdot)\) \(\chi_{927}(724,\cdot)\) \(\chi_{927}(745,\cdot)\) \(\chi_{927}(760,\cdot)\) \(\chi_{927}(763,\cdot)\) \(\chi_{927}(790,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((722,829)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{13}{34}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{25}{51}\right)\)\(e\left(\frac{50}{51}\right)\)\(e\left(\frac{73}{102}\right)\)\(e\left(\frac{10}{51}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{101}{102}\right)\)\(e\left(\frac{44}{51}\right)\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{49}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(106,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 927 }(106,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(106,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 927 }(106,·)) \;\) at \(\; a,b = \) e.g. 1,2