Properties

Label 59.43
Modulus $59$
Conductor $59$
Order $58$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: 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([33]))
 
pari: [g,chi] = znchar(Mod(43,59))
 

Basic properties

Modulus: \(59\)
Conductor: \(59\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 59.d

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

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: Number field defined by a degree 58 polynomial

Values on generators

\(2\) → \(e\left(\frac{33}{58}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 59 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{33}{58}\right)\)\(e\left(\frac{13}{29}\right)\)\(e\left(\frac{4}{29}\right)\)\(e\left(\frac{12}{29}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{41}{58}\right)\)\(e\left(\frac{26}{29}\right)\)\(e\left(\frac{57}{58}\right)\)\(e\left(\frac{13}{58}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 59 }(43,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 59 }(43,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 59 }(43,·),\chi_{ 59 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 59 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2