Properties

Conductor 81
Order 27
Real No
Primitive Yes
Parity Even
Orbit Label 81.g

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(81)
sage: chi = H[31]
pari: [g,chi] = znchar(Mod(31,81))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 81
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 27
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 = 81.g
Orbit index = 7

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 ]

\(\chi_{81}(4,\cdot)\) \(\chi_{81}(7,\cdot)\) \(\chi_{81}(13,\cdot)\) \(\chi_{81}(16,\cdot)\) \(\chi_{81}(22,\cdot)\) \(\chi_{81}(25,\cdot)\) \(\chi_{81}(31,\cdot)\) \(\chi_{81}(34,\cdot)\) \(\chi_{81}(40,\cdot)\) \(\chi_{81}(43,\cdot)\) \(\chi_{81}(49,\cdot)\) \(\chi_{81}(52,\cdot)\) \(\chi_{81}(58,\cdot)\) \(\chi_{81}(61,\cdot)\) \(\chi_{81}(67,\cdot)\) \(\chi_{81}(70,\cdot)\) \(\chi_{81}(76,\cdot)\) \(\chi_{81}(79,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{10}{27}\right)\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{13}{27}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{27})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 81 }(31,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{81}(31,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(31,r) e\left(\frac{2r}{81}\right) = -5.650309122+-7.0052842073i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 81 }(31,·),\chi_{ 81 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{81}(31,\cdot),\chi_{81}(1,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(31,r) \chi_{81}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 81 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{81}(31,·)) = \sum_{r \in \Z/81\Z} \chi_{81}(31,r) e\left(\frac{1 r + 2 r^{-1}}{81}\right) = 4.8027314269+11.1339829279i \)