Properties

Modulus 59
Conductor 59
Order 29
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 59.c

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(59)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20]))
 
pari: [g,chi] = znchar(Mod(17,59))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 59
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
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 59.c
Orbit index = 3

Galois orbit

sage: chi.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

\(2\) → \(e\left(\frac{20}{29}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{20}{29}\right)\)\(e\left(\frac{14}{29}\right)\)\(e\left(\frac{11}{29}\right)\)\(e\left(\frac{4}{29}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{12}{29}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{28}{29}\right)\)\(e\left(\frac{24}{29}\right)\)\(e\left(\frac{7}{29}\right)\)
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 59 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{59}(17,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(17,r) e\left(\frac{2r}{59}\right) = -5.8429166274+4.9860129648i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 59 }(17,·),\chi_{ 59 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{59}(17,\cdot),\chi_{59}(1,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(17,r) \chi_{59}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 59 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{59}(17,·)) = \sum_{r \in \Z/59\Z} \chi_{59}(17,r) e\left(\frac{1 r + 2 r^{-1}}{59}\right) = 7.3050528845+-10.7741469407i \)