Properties

Label 57.2
Modulus $57$
Conductor $57$
Order $18$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(57, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,1]))
 
pari: [g,chi] = znchar(Mod(2,57))
 

Basic properties

Modulus: \(57\)
Conductor: \(57\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
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 57.j

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

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: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{57})^+\)

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{9}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 57 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{57}(2,\cdot)) = \sum_{r\in \Z/57\Z} \chi_{57}(2,r) e\left(\frac{2r}{57}\right) = -4.9856145279+-5.6695368223i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 57 }(2,·),\chi_{ 57 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{57}(2,\cdot),\chi_{57}(1,\cdot)) = \sum_{r\in \Z/57\Z} \chi_{57}(2,r) \chi_{57}(1,1-r) = 1 \)

Kloosterman sum

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