Properties

Label 927.17
Modulus $927$
Conductor $309$
Order $102$
Real no
Primitive no
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([51,70]))
 
pari: [g,chi] = znchar(Mod(17,927))
 

Basic properties

Modulus: \(927\)
Conductor: \(309\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{309}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 927.be

\(\chi_{927}(17,\cdot)\) \(\chi_{927}(26,\cdot)\) \(\chi_{927}(98,\cdot)\) \(\chi_{927}(107,\cdot)\) \(\chi_{927}(152,\cdot)\) \(\chi_{927}(161,\cdot)\) \(\chi_{927}(224,\cdot)\) \(\chi_{927}(242,\cdot)\) \(\chi_{927}(269,\cdot)\) \(\chi_{927}(341,\cdot)\) \(\chi_{927}(350,\cdot)\) \(\chi_{927}(359,\cdot)\) \(\chi_{927}(368,\cdot)\) \(\chi_{927}(377,\cdot)\) \(\chi_{927}(431,\cdot)\) \(\chi_{927}(440,\cdot)\) \(\chi_{927}(467,\cdot)\) \(\chi_{927}(494,\cdot)\) \(\chi_{927}(503,\cdot)\) \(\chi_{927}(530,\cdot)\) \(\chi_{927}(548,\cdot)\) \(\chi_{927}(575,\cdot)\) \(\chi_{927}(620,\cdot)\) \(\chi_{927}(647,\cdot)\) \(\chi_{927}(656,\cdot)\) \(\chi_{927}(701,\cdot)\) \(\chi_{927}(710,\cdot)\) \(\chi_{927}(728,\cdot)\) \(\chi_{927}(737,\cdot)\) \(\chi_{927}(746,\cdot)\) ...

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

\((722,829)\) → \((-1,e\left(\frac{35}{51}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{71}{102}\right)\)\(e\left(\frac{20}{51}\right)\)\(e\left(\frac{19}{102}\right)\)\(e\left(\frac{38}{51}\right)\)\(e\left(\frac{3}{34}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{37}{102}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{40}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 927 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 927 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 927 }(17,·),\chi_{ 927 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 927 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2