Properties

Conductor 569
Order 71
Real No
Primitive Yes
Parity Even
Orbit Label 569.e

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(569)
sage: chi = H[16]
pari: [g,chi] = znchar(Mod(16,569))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 569
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 71
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 = 569.e
Orbit index = 5

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 ]

\(\chi_{569}(5,\cdot)\) \(\chi_{569}(16,\cdot)\) \(\chi_{569}(18,\cdot)\) \(\chi_{569}(25,\cdot)\) \(\chi_{569}(33,\cdot)\) \(\chi_{569}(56,\cdot)\) \(\chi_{569}(63,\cdot)\) \(\chi_{569}(69,\cdot)\) \(\chi_{569}(80,\cdot)\) \(\chi_{569}(90,\cdot)\) \(\chi_{569}(101,\cdot)\) \(\chi_{569}(104,\cdot)\) \(\chi_{569}(107,\cdot)\) \(\chi_{569}(111,\cdot)\) \(\chi_{569}(113,\cdot)\) \(\chi_{569}(114,\cdot)\) \(\chi_{569}(117,\cdot)\) \(\chi_{569}(125,\cdot)\) \(\chi_{569}(134,\cdot)\) \(\chi_{569}(136,\cdot)\) \(\chi_{569}(141,\cdot)\) \(\chi_{569}(142,\cdot)\) \(\chi_{569}(153,\cdot)\) \(\chi_{569}(164,\cdot)\) \(\chi_{569}(165,\cdot)\) \(\chi_{569}(172,\cdot)\) \(\chi_{569}(196,\cdot)\) \(\chi_{569}(209,\cdot)\) \(\chi_{569}(219,\cdot)\) \(\chi_{569}(249,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{42}{71}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{49}{71}\right)\)\(e\left(\frac{42}{71}\right)\)\(e\left(\frac{27}{71}\right)\)\(e\left(\frac{7}{71}\right)\)\(e\left(\frac{20}{71}\right)\)\(e\left(\frac{23}{71}\right)\)\(e\left(\frac{5}{71}\right)\)\(e\left(\frac{13}{71}\right)\)\(e\left(\frac{56}{71}\right)\)\(e\left(\frac{52}{71}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{71})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 569 }(16,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{569}(16,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(16,r) e\left(\frac{2r}{569}\right) = -16.1064815691+17.5949211838i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 569 }(16,·),\chi_{ 569 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{569}(16,\cdot),\chi_{569}(1,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(16,r) \chi_{569}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 569 }(16,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{569}(16,·)) = \sum_{r \in \Z/569\Z} \chi_{569}(16,r) e\left(\frac{1 r + 2 r^{-1}}{569}\right) = 13.0972808731+-19.2537324631i \)