Properties

Conductor 59
Order 29
Real No
Primitive Yes
Parity Even
Orbit Label 59.c

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[35]
pari: [g,chi] = znchar(Mod(35,59))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 59
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 29
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 = 59.c
Orbit index = 3

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 ]

\(\chi_{59}(3,\cdot)\) \(\chi_{59}(4,\cdot)\) \(\chi_{59}(5,\cdot)\) \(\chi_{59}(7,\cdot)\) \(\chi_{59}(9,\cdot)\) \(\chi_{59}(12,\cdot)\) \(\chi_{59}(15,\cdot)\) \(\chi_{59}(16,\cdot)\) \(\chi_{59}(17,\cdot)\) \(\chi_{59}(19,\cdot)\) \(\chi_{59}(20,\cdot)\) \(\chi_{59}(21,\cdot)\) \(\chi_{59}(22,\cdot)\) \(\chi_{59}(25,\cdot)\) \(\chi_{59}(26,\cdot)\) \(\chi_{59}(27,\cdot)\) \(\chi_{59}(28,\cdot)\) \(\chi_{59}(29,\cdot)\) \(\chi_{59}(35,\cdot)\) \(\chi_{59}(36,\cdot)\) \(\chi_{59}(41,\cdot)\) \(\chi_{59}(45,\cdot)\) \(\chi_{59}(46,\cdot)\) \(\chi_{59}(48,\cdot)\) \(\chi_{59}(49,\cdot)\) \(\chi_{59}(51,\cdot)\) \(\chi_{59}(53,\cdot)\) \(\chi_{59}(57,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{12}{29}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{12}{29}\right)\)\(e\left(\frac{20}{29}\right)\)\(e\left(\frac{24}{29}\right)\)\(e\left(\frac{14}{29}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{13}{29}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{11}{29}\right)\)\(e\left(\frac{26}{29}\right)\)\(e\left(\frac{10}{29}\right)\)
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)
\( \tau_{ a }( \chi_{ 59 }(35,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{59}(35,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(35,r) e\left(\frac{2r}{59}\right) = 6.4555324222+4.1624633508i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 59 }(35,·),\chi_{ 59 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{59}(35,\cdot),\chi_{59}(1,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(35,r) \chi_{59}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 59 }(35,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{59}(35,·)) = \sum_{r \in \Z/59\Z} \chi_{59}(35,r) e\left(\frac{1 r + 2 r^{-1}}{59}\right) = -0.3752523891+-1.3515369574i \)