Properties

Label 927.734
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([85,72]))
 
pari: [g,chi] = znchar(Mod(734,927))
 

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.bf

\(\chi_{927}(14,\cdot)\) \(\chi_{927}(23,\cdot)\) \(\chi_{927}(137,\cdot)\) \(\chi_{927}(164,\cdot)\) \(\chi_{927}(167,\cdot)\) \(\chi_{927}(182,\cdot)\) \(\chi_{927}(203,\cdot)\) \(\chi_{927}(236,\cdot)\) \(\chi_{927}(272,\cdot)\) \(\chi_{927}(299,\cdot)\) \(\chi_{927}(317,\cdot)\) \(\chi_{927}(425,\cdot)\) \(\chi_{927}(446,\cdot)\) \(\chi_{927}(473,\cdot)\) \(\chi_{927}(488,\cdot)\) \(\chi_{927}(491,\cdot)\) \(\chi_{927}(524,\cdot)\) \(\chi_{927}(545,\cdot)\) \(\chi_{927}(581,\cdot)\) \(\chi_{927}(587,\cdot)\) \(\chi_{927}(596,\cdot)\) \(\chi_{927}(608,\cdot)\) \(\chi_{927}(626,\cdot)\) \(\chi_{927}(632,\cdot)\) \(\chi_{927}(641,\cdot)\) \(\chi_{927}(734,\cdot)\) \(\chi_{927}(785,\cdot)\) \(\chi_{927}(797,\cdot)\) \(\chi_{927}(821,\cdot)\) \(\chi_{927}(833,\cdot)\) ...

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

\((722,829)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{12}{17}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{91}{102}\right)\)\(e\left(\frac{40}{51}\right)\)\(e\left(\frac{89}{102}\right)\)\(e\left(\frac{8}{51}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{91}{102}\right)\)\(e\left(\frac{25}{51}\right)\)\(e\left(\frac{5}{102}\right)\)\(e\left(\frac{29}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(734,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 927 }(734,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(734,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 927 }(734,·)) \;\) at \(\; a,b = \) e.g. 1,2