Properties

Label 243.32
Modulus $243$
Conductor $243$
Order $162$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(162))
 
M = H._module
 
chi = DirichletCharacter(H, M([5]))
 
pari: [g,chi] = znchar(Mod(32,243))
 

Basic properties

Modulus: \(243\)
Conductor: \(243\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(162\)
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 243.j

\(\chi_{243}(2,\cdot)\) \(\chi_{243}(5,\cdot)\) \(\chi_{243}(11,\cdot)\) \(\chi_{243}(14,\cdot)\) \(\chi_{243}(20,\cdot)\) \(\chi_{243}(23,\cdot)\) \(\chi_{243}(29,\cdot)\) \(\chi_{243}(32,\cdot)\) \(\chi_{243}(38,\cdot)\) \(\chi_{243}(41,\cdot)\) \(\chi_{243}(47,\cdot)\) \(\chi_{243}(50,\cdot)\) \(\chi_{243}(56,\cdot)\) \(\chi_{243}(59,\cdot)\) \(\chi_{243}(65,\cdot)\) \(\chi_{243}(68,\cdot)\) \(\chi_{243}(74,\cdot)\) \(\chi_{243}(77,\cdot)\) \(\chi_{243}(83,\cdot)\) \(\chi_{243}(86,\cdot)\) \(\chi_{243}(92,\cdot)\) \(\chi_{243}(95,\cdot)\) \(\chi_{243}(101,\cdot)\) \(\chi_{243}(104,\cdot)\) \(\chi_{243}(110,\cdot)\) \(\chi_{243}(113,\cdot)\) \(\chi_{243}(119,\cdot)\) \(\chi_{243}(122,\cdot)\) \(\chi_{243}(128,\cdot)\) \(\chi_{243}(131,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{81})$
Fixed field: Number field defined by a degree 162 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{5}{162}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 243 }(32, a) \) \(-1\)\(1\)\(e\left(\frac{5}{162}\right)\)\(e\left(\frac{5}{81}\right)\)\(e\left(\frac{115}{162}\right)\)\(e\left(\frac{13}{81}\right)\)\(e\left(\frac{5}{54}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{119}{162}\right)\)\(e\left(\frac{20}{81}\right)\)\(e\left(\frac{31}{162}\right)\)\(e\left(\frac{10}{81}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 243 }(32,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 243 }(32,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 243 }(32,·),\chi_{ 243 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 243 }(32,·)) \;\) at \(\; a,b = \) e.g. 1,2