Properties

Conductor 59
Order 58
Real No
Primitive Yes
Parity Odd
Orbit Label 59.d

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(59)
sage: chi = H[11]
pari: [g,chi] = znchar(Mod(11,59))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 59
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 58
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 = 59.d
Orbit index = 4

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 ]

\(\chi_{59}(2,\cdot)\) \(\chi_{59}(6,\cdot)\) \(\chi_{59}(8,\cdot)\) \(\chi_{59}(10,\cdot)\) \(\chi_{59}(11,\cdot)\) \(\chi_{59}(13,\cdot)\) \(\chi_{59}(14,\cdot)\) \(\chi_{59}(18,\cdot)\) \(\chi_{59}(23,\cdot)\) \(\chi_{59}(24,\cdot)\) \(\chi_{59}(30,\cdot)\) \(\chi_{59}(31,\cdot)\) \(\chi_{59}(32,\cdot)\) \(\chi_{59}(33,\cdot)\) \(\chi_{59}(34,\cdot)\) \(\chi_{59}(37,\cdot)\) \(\chi_{59}(38,\cdot)\) \(\chi_{59}(39,\cdot)\) \(\chi_{59}(40,\cdot)\) \(\chi_{59}(42,\cdot)\) \(\chi_{59}(43,\cdot)\) \(\chi_{59}(44,\cdot)\) \(\chi_{59}(47,\cdot)\) \(\chi_{59}(50,\cdot)\) \(\chi_{59}(52,\cdot)\) \(\chi_{59}(54,\cdot)\) \(\chi_{59}(55,\cdot)\) \(\chi_{59}(56,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{25}{58}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{25}{58}\right)\)\(e\left(\frac{16}{29}\right)\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{17}{29}\right)\)\(e\left(\frac{57}{58}\right)\)\(e\left(\frac{22}{29}\right)\)\(e\left(\frac{17}{58}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{45}{58}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{29})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 59 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{59}(11,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(11,r) e\left(\frac{2r}{59}\right) = 4.7138535441+6.0646174459i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 59 }(11,·),\chi_{ 59 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{59}(11,\cdot),\chi_{59}(1,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(11,r) \chi_{59}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 59 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{59}(11,·)) = \sum_{r \in \Z/59\Z} \chi_{59}(11,r) e\left(\frac{1 r + 2 r^{-1}}{59}\right) = -8.6474985825+1.9034583761i \)