Properties

Label 432.59
Modulus $432$
Conductor $432$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(432, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([18,9,10]))
 
pari: [g,chi] = znchar(Mod(59,432))
 

Basic properties

Modulus: \(432\)
Conductor: \(432\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 432.bj

\(\chi_{432}(11,\cdot)\) \(\chi_{432}(59,\cdot)\) \(\chi_{432}(83,\cdot)\) \(\chi_{432}(131,\cdot)\) \(\chi_{432}(155,\cdot)\) \(\chi_{432}(203,\cdot)\) \(\chi_{432}(227,\cdot)\) \(\chi_{432}(275,\cdot)\) \(\chi_{432}(299,\cdot)\) \(\chi_{432}(347,\cdot)\) \(\chi_{432}(371,\cdot)\) \(\chi_{432}(419,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.5532004127928253705369187176396364210546696053048780432717505515499814912.1

Values on generators

\((271,325,353)\) → \((-1,i,e\left(\frac{5}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{1}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 432 }(59,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 432 }(59,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 432 }(59,·),\chi_{ 432 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 432 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2