Properties

Modulus 137
Conductor 137
Order 8
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 137.d

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(137)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5]))
 
pari: [g,chi] = znchar(Mod(10,137))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 137
Conductor = 137
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
Real = no
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 = 137.d
Orbit index = 4

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 ]
 

\(\chi_{137}(10,\cdot)\) \(\chi_{137}(41,\cdot)\) \(\chi_{137}(96,\cdot)\) \(\chi_{137}(127,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{5}{8}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(i\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(i\)\(-i\)\(i\)\(e\left(\frac{1}{8}\right)\)\(i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 137 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{137}(10,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(10,r) e\left(\frac{2r}{137}\right) = 7.918154715+-8.619908695i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 137 }(10,·),\chi_{ 137 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{137}(10,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(10,r) \chi_{137}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 137 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{137}(10,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(10,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = -14.040646051+14.040646051i \)