Properties

Label 432.85
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([0,9,4]))
 
pari: [g,chi] = znchar(Mod(85,432))
 

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.bg

\(\chi_{432}(13,\cdot)\) \(\chi_{432}(61,\cdot)\) \(\chi_{432}(85,\cdot)\) \(\chi_{432}(133,\cdot)\) \(\chi_{432}(157,\cdot)\) \(\chi_{432}(205,\cdot)\) \(\chi_{432}(229,\cdot)\) \(\chi_{432}(277,\cdot)\) \(\chi_{432}(301,\cdot)\) \(\chi_{432}(349,\cdot)\) \(\chi_{432}(373,\cdot)\) \(\chi_{432}(421,\cdot)\)

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: 36.36.614667125325361522818798575155151578949632894783197825857500612833312768.1

Values on generators

\((271,325,353)\) → \((1,i,e\left(\frac{1}{9}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 432 }(85,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 432 }(85,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 432 }(85,·),\chi_{ 432 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 432 }(85,·)) \;\) at \(\; a,b = \) e.g. 1,2