Properties

Label 927.2
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([17,44]))
 
pari: [g,chi] = znchar(Mod(2,927))
 

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.bn

\(\chi_{927}(2,\cdot)\) \(\chi_{927}(29,\cdot)\) \(\chi_{927}(32,\cdot)\) \(\chi_{927}(50,\cdot)\) \(\chi_{927}(83,\cdot)\) \(\chi_{927}(119,\cdot)\) \(\chi_{927}(128,\cdot)\) \(\chi_{927}(185,\cdot)\) \(\chi_{927}(194,\cdot)\) \(\chi_{927}(200,\cdot)\) \(\chi_{927}(221,\cdot)\) \(\chi_{927}(239,\cdot)\) \(\chi_{927}(335,\cdot)\) \(\chi_{927}(347,\cdot)\) \(\chi_{927}(401,\cdot)\) \(\chi_{927}(419,\cdot)\) \(\chi_{927}(461,\cdot)\) \(\chi_{927}(464,\cdot)\) \(\chi_{927}(533,\cdot)\) \(\chi_{927}(635,\cdot)\) \(\chi_{927}(659,\cdot)\) \(\chi_{927}(677,\cdot)\) \(\chi_{927}(686,\cdot)\) \(\chi_{927}(716,\cdot)\) \(\chi_{927}(725,\cdot)\) \(\chi_{927}(740,\cdot)\) \(\chi_{927}(749,\cdot)\) \(\chi_{927}(776,\cdot)\) \(\chi_{927}(779,\cdot)\) \(\chi_{927}(860,\cdot)\) ...

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

\((722,829)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{22}{51}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{5}{34}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{20}{51}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{49}{102}\right)\)\(e\left(\frac{20}{51}\right)\)\(e\left(\frac{55}{102}\right)\)\(e\left(\frac{10}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

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

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(2,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

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