Properties

Modulus 57
Conductor 57
Order 18
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 57.j

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(57)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,7]))
 
pari: [g,chi] = znchar(Mod(14,57))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 57
Conductor = 57
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 18
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 = even
Orbit label = 57.j
Orbit index = 10

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 ]
 

\(\chi_{57}(2,\cdot)\) \(\chi_{57}(14,\cdot)\) \(\chi_{57}(29,\cdot)\) \(\chi_{57}(32,\cdot)\) \(\chi_{57}(41,\cdot)\) \(\chi_{57}(53,\cdot)\)

Values on generators

\((20,40)\) → \((-1,e\left(\frac{7}{18}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{9}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{9})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 57 }(14,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{57}(14,\cdot)) = \sum_{r\in \Z/57\Z} \chi_{57}(14,r) e\left(\frac{2r}{57}\right) = 6.3245521672+-4.1231104623i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 57 }(14,·),\chi_{ 57 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{57}(14,\cdot),\chi_{57}(1,\cdot)) = \sum_{r\in \Z/57\Z} \chi_{57}(14,r) \chi_{57}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 57 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{57}(14,·)) = \sum_{r \in \Z/57\Z} \chi_{57}(14,r) e\left(\frac{1 r + 2 r^{-1}}{57}\right) = 0.0 \)