Properties

Label 59.21
Modulus $59$
Conductor $59$
Order $29$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(59, base_ring=CyclotomicField(58))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10]))
 
pari: [g,chi] = znchar(Mod(21,59))
 

Basic properties

Modulus: \(59\)
Conductor: \(59\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(29\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 59.c

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

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{29})$
Fixed field: \(\Q(\zeta_{59})^+\)

Values on generators

\(2\) → \(e\left(\frac{5}{29}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{18}{29}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{1}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{15}{29}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{6}{29}\right)\)\(e\left(\frac{9}{29}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 59 }(21,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{59}(21,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(21,r) e\left(\frac{2r}{59}\right) = 7.6690539698+0.4308261936i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 59 }(21,·),\chi_{ 59 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{59}(21,\cdot),\chi_{59}(1,\cdot)) = \sum_{r\in \Z/59\Z} \chi_{59}(21,r) \chi_{59}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 59 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{59}(21,·)) = \sum_{r \in \Z/59\Z} \chi_{59}(21,r) e\left(\frac{1 r + 2 r^{-1}}{59}\right) = -8.4582953142+-5.0891874348i \)