Properties

Label 729.2
Modulus $729$
Conductor $729$
Order $486$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(486))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(2,729))
 

Basic properties

Modulus: \(729\)
Conductor: \(729\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(486\)
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 729.l

\(\chi_{729}(2,\cdot)\) \(\chi_{729}(5,\cdot)\) \(\chi_{729}(11,\cdot)\) \(\chi_{729}(14,\cdot)\) \(\chi_{729}(20,\cdot)\) \(\chi_{729}(23,\cdot)\) \(\chi_{729}(29,\cdot)\) \(\chi_{729}(32,\cdot)\) \(\chi_{729}(38,\cdot)\) \(\chi_{729}(41,\cdot)\) \(\chi_{729}(47,\cdot)\) \(\chi_{729}(50,\cdot)\) \(\chi_{729}(56,\cdot)\) \(\chi_{729}(59,\cdot)\) \(\chi_{729}(65,\cdot)\) \(\chi_{729}(68,\cdot)\) \(\chi_{729}(74,\cdot)\) \(\chi_{729}(77,\cdot)\) \(\chi_{729}(83,\cdot)\) \(\chi_{729}(86,\cdot)\) \(\chi_{729}(92,\cdot)\) \(\chi_{729}(95,\cdot)\) \(\chi_{729}(101,\cdot)\) \(\chi_{729}(104,\cdot)\) \(\chi_{729}(110,\cdot)\) \(\chi_{729}(113,\cdot)\) \(\chi_{729}(119,\cdot)\) \(\chi_{729}(122,\cdot)\) \(\chi_{729}(128,\cdot)\) \(\chi_{729}(131,\cdot)\) ...

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: $\Q(\zeta_{243})$
Fixed field: Number field defined by a degree 486 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1}{486}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 729 }(2, a) \) \(-1\)\(1\)\(e\left(\frac{1}{486}\right)\)\(e\left(\frac{1}{243}\right)\)\(e\left(\frac{23}{486}\right)\)\(e\left(\frac{197}{243}\right)\)\(e\left(\frac{1}{162}\right)\)\(e\left(\frac{4}{81}\right)\)\(e\left(\frac{283}{486}\right)\)\(e\left(\frac{166}{243}\right)\)\(e\left(\frac{395}{486}\right)\)\(e\left(\frac{2}{243}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 729 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 729 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 729 }(2,·),\chi_{ 729 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 729 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2