Properties

Conductor 57
Order 18
Real No
Primitive Yes
Parity Even
Orbit Label 57.j

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(57)
sage: chi = H[53]
pari: [g,chi] = znchar(Mod(53,57))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
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
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 57.j
Orbit index = 10

Galois orbit

sage: chi.sage_character().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

\((20,40)\) → \((-1,e\left(\frac{11}{18}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{4}{9}\right)\)
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 57 }(53,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{57}(53,\cdot)) = \sum_{r\in \Z/57\Z} \chi_{57}(53,r) e\left(\frac{2r}{57}\right) = 6.3245521672+4.1231104623i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 57 }(53,·),\chi_{ 57 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{57}(53,\cdot),\chi_{57}(1,\cdot)) = \sum_{r\in \Z/57\Z} \chi_{57}(53,r) \chi_{57}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 57 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{57}(53,·)) = \sum_{r \in \Z/57\Z} \chi_{57}(53,r) e\left(\frac{1 r + 2 r^{-1}}{57}\right) = 0.0 \)