Properties

Conductor 569
Order 142
Real No
Primitive Yes
Parity Even
Orbit Label 569.f

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[26]
pari: [g,chi] = znchar(Mod(26,569))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 569
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 142
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.f
Orbit index = 6

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 ]

\(\chi_{569}(4,\cdot)\) \(\chi_{569}(14,\cdot)\) \(\chi_{569}(20,\cdot)\) \(\chi_{569}(26,\cdot)\) \(\chi_{569}(34,\cdot)\) \(\chi_{569}(41,\cdot)\) \(\chi_{569}(43,\cdot)\) \(\chi_{569}(49,\cdot)\) \(\chi_{569}(64,\cdot)\) \(\chi_{569}(70,\cdot)\) \(\chi_{569}(72,\cdot)\) \(\chi_{569}(81,\cdot)\) \(\chi_{569}(87,\cdot)\) \(\chi_{569}(91,\cdot)\) \(\chi_{569}(93,\cdot)\) \(\chi_{569}(100,\cdot)\) \(\chi_{569}(119,\cdot)\) \(\chi_{569}(122,\cdot)\) \(\chi_{569}(130,\cdot)\) \(\chi_{569}(132,\cdot)\) \(\chi_{569}(158,\cdot)\) \(\chi_{569}(169,\cdot)\) \(\chi_{569}(170,\cdot)\) \(\chi_{569}(197,\cdot)\) \(\chi_{569}(205,\cdot)\) \(\chi_{569}(215,\cdot)\) \(\chi_{569}(221,\cdot)\) \(\chi_{569}(224,\cdot)\) \(\chi_{569}(242,\cdot)\) \(\chi_{569}(245,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{137}{142}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{3}{71}\right)\)\(e\left(\frac{137}{142}\right)\)\(e\left(\frac{6}{71}\right)\)\(e\left(\frac{41}{71}\right)\)\(e\left(\frac{1}{142}\right)\)\(e\left(\frac{13}{71}\right)\)\(e\left(\frac{9}{71}\right)\)\(e\left(\frac{66}{71}\right)\)\(e\left(\frac{44}{71}\right)\)\(e\left(\frac{31}{142}\right)\)
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)
\( \tau_{ a }( \chi_{ 569 }(26,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{569}(26,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(26,r) e\left(\frac{2r}{569}\right) = 22.9270228+6.5841951314i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 569 }(26,·),\chi_{ 569 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{569}(26,\cdot),\chi_{569}(1,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(26,r) \chi_{569}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 569 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{569}(26,·)) = \sum_{r \in \Z/569\Z} \chi_{569}(26,r) e\left(\frac{1 r + 2 r^{-1}}{569}\right) = -5.2520829507+-0.7013031139i \)