Properties

Label 51.29
Modulus $51$
Conductor $51$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(51, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8,13]))
 
pari: [g,chi] = znchar(Mod(29,51))
 

Basic properties

Modulus: \(51\)
Conductor: \(51\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
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 51.i

\(\chi_{51}(5,\cdot)\) \(\chi_{51}(11,\cdot)\) \(\chi_{51}(14,\cdot)\) \(\chi_{51}(20,\cdot)\) \(\chi_{51}(23,\cdot)\) \(\chi_{51}(29,\cdot)\) \(\chi_{51}(41,\cdot)\) \(\chi_{51}(44,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{51})^+\)

Values on generators

\((35,37)\) → \((-1,e\left(\frac{13}{16}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(i\)\(e\left(\frac{13}{16}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 51 }(29,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 51 }(29,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 51 }(29,·),\chi_{ 51 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 51 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2