Properties

Conductor 163
Order 81
Real No
Primitive Yes
Parity Even
Orbit Label 163.i

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(163)
sage: chi = H[39]
pari: [g,chi] = znchar(Mod(39,163))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 163
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 81
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 = 163.i
Orbit index = 9

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 ]

\(\chi_{163}(4,\cdot)\) \(\chi_{163}(9,\cdot)\) \(\chi_{163}(10,\cdot)\) \(\chi_{163}(14,\cdot)\) \(\chi_{163}(15,\cdot)\) \(\chi_{163}(16,\cdot)\) \(\chi_{163}(24,\cdot)\) \(\chi_{163}(26,\cdot)\) \(\chi_{163}(33,\cdot)\) \(\chi_{163}(34,\cdot)\) \(\chi_{163}(35,\cdot)\) \(\chi_{163}(39,\cdot)\) \(\chi_{163}(41,\cdot)\) \(\chi_{163}(43,\cdot)\) \(\chi_{163}(46,\cdot)\) \(\chi_{163}(47,\cdot)\) \(\chi_{163}(49,\cdot)\) \(\chi_{163}(51,\cdot)\) \(\chi_{163}(54,\cdot)\) \(\chi_{163}(55,\cdot)\) \(\chi_{163}(56,\cdot)\) \(\chi_{163}(57,\cdot)\) \(\chi_{163}(60,\cdot)\) \(\chi_{163}(62,\cdot)\) \(\chi_{163}(69,\cdot)\) \(\chi_{163}(71,\cdot)\) \(\chi_{163}(74,\cdot)\) \(\chi_{163}(81,\cdot)\) \(\chi_{163}(83,\cdot)\) \(\chi_{163}(84,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{76}{81}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{76}{81}\right)\)\(e\left(\frac{62}{81}\right)\)\(e\left(\frac{71}{81}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{40}{81}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{43}{81}\right)\)\(e\left(\frac{1}{81}\right)\)\(e\left(\frac{8}{81}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{81})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 163 }(39,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{163}(39,\cdot)) = \sum_{r\in \Z/163\Z} \chi_{163}(39,r) e\left(\frac{2r}{163}\right) = 6.4899005619+-10.9945982508i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 163 }(39,·),\chi_{ 163 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{163}(39,\cdot),\chi_{163}(1,\cdot)) = \sum_{r\in \Z/163\Z} \chi_{163}(39,r) \chi_{163}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 163 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{163}(39,·)) = \sum_{r \in \Z/163\Z} \chi_{163}(39,r) e\left(\frac{1 r + 2 r^{-1}}{163}\right) = 13.733859629+-2.6972421005i \)