Properties

Conductor 709
Order 59
Real No
Primitive Yes
Parity Even
Orbit Label 709.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(709)
sage: chi = H[339]
pari: [g,chi] = znchar(Mod(339,709))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 709
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 59
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 = 709.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 ]

\(\chi_{709}(20,\cdot)\) \(\chi_{709}(27,\cdot)\) \(\chi_{709}(44,\cdot)\) \(\chi_{709}(59,\cdot)\) \(\chi_{709}(63,\cdot)\) \(\chi_{709}(75,\cdot)\) \(\chi_{709}(82,\cdot)\) \(\chi_{709}(87,\cdot)\) \(\chi_{709}(104,\cdot)\) \(\chi_{709}(138,\cdot)\) \(\chi_{709}(144,\cdot)\) \(\chi_{709}(147,\cdot)\) \(\chi_{709}(149,\cdot)\) \(\chi_{709}(163,\cdot)\) \(\chi_{709}(165,\cdot)\) \(\chi_{709}(170,\cdot)\) \(\chi_{709}(171,\cdot)\) \(\chi_{709}(172,\cdot)\) \(\chi_{709}(175,\cdot)\) \(\chi_{709}(181,\cdot)\) \(\chi_{709}(186,\cdot)\) \(\chi_{709}(201,\cdot)\) \(\chi_{709}(203,\cdot)\) \(\chi_{709}(222,\cdot)\) \(\chi_{709}(283,\cdot)\) \(\chi_{709}(322,\cdot)\) \(\chi_{709}(336,\cdot)\) \(\chi_{709}(339,\cdot)\) \(\chi_{709}(343,\cdot)\) \(\chi_{709}(363,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{46}{59}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{46}{59}\right)\)\(e\left(\frac{6}{59}\right)\)\(e\left(\frac{33}{59}\right)\)\(e\left(\frac{3}{59}\right)\)\(e\left(\frac{52}{59}\right)\)\(e\left(\frac{32}{59}\right)\)\(e\left(\frac{20}{59}\right)\)\(e\left(\frac{12}{59}\right)\)\(e\left(\frac{49}{59}\right)\)\(e\left(\frac{22}{59}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{59})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 709 }(339,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{709}(339,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(339,r) e\left(\frac{2r}{709}\right) = -23.3489868947+12.7994066656i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 709 }(339,·),\chi_{ 709 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{709}(339,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(339,r) \chi_{709}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 709 }(339,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{709}(339,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(339,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = -10.1995207045+8.4560891124i \)