Properties

Modulus 127
Conductor 127
Order 2
Real yes
Primitive yes
Minimal yes
Parity odd
Orbit label 127.b

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(127)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(126,127))
 

Kronecker symbol representation

sage: kronecker_character(-127)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-127}{\bullet}\right)\)

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 127
Conductor = 127
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 2
Real = yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 127.b
Orbit index = 2

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{127}(126,\cdot)\)

Values on generators

\(3\) → \(-1\)

Values

-11234567891011
\(-1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 127 }(126,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{127}(126,\cdot)) = \sum_{r\in \Z/127\Z} \chi_{127}(126,r) e\left(\frac{2r}{127}\right) = 11.2694276696i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 127 }(126,·),\chi_{ 127 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{127}(126,\cdot),\chi_{127}(1,\cdot)) = \sum_{r\in \Z/127\Z} \chi_{127}(126,r) \chi_{127}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 127 }(126,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{127}(126,·)) = \sum_{r \in \Z/127\Z} \chi_{127}(126,r) e\left(\frac{1 r + 2 r^{-1}}{127}\right) = -0.2787641615i \)