Properties

Label 729.8
Modulus $729$
Conductor $243$
Order $162$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(162))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(8,729))
 

Basic properties

Modulus: \(729\)
Conductor: \(243\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(162\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{243}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 729.j

\(\chi_{729}(8,\cdot)\) \(\chi_{729}(17,\cdot)\) \(\chi_{729}(35,\cdot)\) \(\chi_{729}(44,\cdot)\) \(\chi_{729}(62,\cdot)\) \(\chi_{729}(71,\cdot)\) \(\chi_{729}(89,\cdot)\) \(\chi_{729}(98,\cdot)\) \(\chi_{729}(116,\cdot)\) \(\chi_{729}(125,\cdot)\) \(\chi_{729}(143,\cdot)\) \(\chi_{729}(152,\cdot)\) \(\chi_{729}(170,\cdot)\) \(\chi_{729}(179,\cdot)\) \(\chi_{729}(197,\cdot)\) \(\chi_{729}(206,\cdot)\) \(\chi_{729}(224,\cdot)\) \(\chi_{729}(233,\cdot)\) \(\chi_{729}(251,\cdot)\) \(\chi_{729}(260,\cdot)\) \(\chi_{729}(278,\cdot)\) \(\chi_{729}(287,\cdot)\) \(\chi_{729}(305,\cdot)\) \(\chi_{729}(314,\cdot)\) \(\chi_{729}(332,\cdot)\) \(\chi_{729}(341,\cdot)\) \(\chi_{729}(359,\cdot)\) \(\chi_{729}(368,\cdot)\) \(\chi_{729}(386,\cdot)\) \(\chi_{729}(395,\cdot)\) ...

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

\(2\) → \(e\left(\frac{1}{162}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 729 }(8, a) \) \(-1\)\(1\)\(e\left(\frac{1}{162}\right)\)\(e\left(\frac{1}{81}\right)\)\(e\left(\frac{23}{162}\right)\)\(e\left(\frac{35}{81}\right)\)\(e\left(\frac{1}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{121}{162}\right)\)\(e\left(\frac{4}{81}\right)\)\(e\left(\frac{71}{162}\right)\)\(e\left(\frac{2}{81}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 729 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 729 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 729 }(8,·),\chi_{ 729 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 729 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2