Properties

Conductor 569
Order 568
Real No
Primitive Yes
Parity Odd
Orbit Label 569.h

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[3]
pari: [g,chi] = znchar(Mod(3,569))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 569
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 568
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 = Odd
Orbit label = 569.h
Orbit index = 8

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 ]

\(\chi_{569}(3,\cdot)\) \(\chi_{569}(6,\cdot)\) \(\chi_{569}(11,\cdot)\) \(\chi_{569}(12,\cdot)\) \(\chi_{569}(15,\cdot)\) \(\chi_{569}(19,\cdot)\) \(\chi_{569}(21,\cdot)\) \(\chi_{569}(22,\cdot)\) \(\chi_{569}(23,\cdot)\) \(\chi_{569}(24,\cdot)\) \(\chi_{569}(27,\cdot)\) \(\chi_{569}(29,\cdot)\) \(\chi_{569}(30,\cdot)\) \(\chi_{569}(31,\cdot)\) \(\chi_{569}(37,\cdot)\) \(\chi_{569}(38,\cdot)\) \(\chi_{569}(39,\cdot)\) \(\chi_{569}(42,\cdot)\) \(\chi_{569}(44,\cdot)\) \(\chi_{569}(46,\cdot)\) \(\chi_{569}(47,\cdot)\) \(\chi_{569}(48,\cdot)\) \(\chi_{569}(51,\cdot)\) \(\chi_{569}(53,\cdot)\) \(\chi_{569}(54,\cdot)\) \(\chi_{569}(55,\cdot)\) \(\chi_{569}(58,\cdot)\) \(\chi_{569}(59,\cdot)\) \(\chi_{569}(60,\cdot)\) \(\chi_{569}(62,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{1}{568}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{255}{284}\right)\)\(e\left(\frac{1}{568}\right)\)\(e\left(\frac{113}{142}\right)\)\(e\left(\frac{37}{71}\right)\)\(e\left(\frac{511}{568}\right)\)\(e\left(\frac{111}{284}\right)\)\(e\left(\frac{197}{284}\right)\)\(e\left(\frac{1}{284}\right)\)\(e\left(\frac{119}{284}\right)\)\(e\left(\frac{79}{568}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{568})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 569 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{569}(3,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(3,r) e\left(\frac{2r}{569}\right) = 9.8367121717+21.7310628744i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 569 }(3,·),\chi_{ 569 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{569}(3,\cdot),\chi_{569}(1,\cdot)) = \sum_{r\in \Z/569\Z} \chi_{569}(3,r) \chi_{569}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 569 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{569}(3,·)) = \sum_{r \in \Z/569\Z} \chi_{569}(3,r) e\left(\frac{1 r + 2 r^{-1}}{569}\right) = -6.5841361523+-19.8154094639i \)