Properties

Label 729.316
Modulus $729$
Conductor $243$
Order $81$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(162))
 
M = H._module
 
chi = DirichletCharacter(H, M([46]))
 
pari: [g,chi] = znchar(Mod(316,729))
 

Basic properties

Modulus: \(729\)
Conductor: \(243\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(81\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{243}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 729.i

\(\chi_{729}(10,\cdot)\) \(\chi_{729}(19,\cdot)\) \(\chi_{729}(37,\cdot)\) \(\chi_{729}(46,\cdot)\) \(\chi_{729}(64,\cdot)\) \(\chi_{729}(73,\cdot)\) \(\chi_{729}(91,\cdot)\) \(\chi_{729}(100,\cdot)\) \(\chi_{729}(118,\cdot)\) \(\chi_{729}(127,\cdot)\) \(\chi_{729}(145,\cdot)\) \(\chi_{729}(154,\cdot)\) \(\chi_{729}(172,\cdot)\) \(\chi_{729}(181,\cdot)\) \(\chi_{729}(199,\cdot)\) \(\chi_{729}(208,\cdot)\) \(\chi_{729}(226,\cdot)\) \(\chi_{729}(235,\cdot)\) \(\chi_{729}(253,\cdot)\) \(\chi_{729}(262,\cdot)\) \(\chi_{729}(280,\cdot)\) \(\chi_{729}(289,\cdot)\) \(\chi_{729}(307,\cdot)\) \(\chi_{729}(316,\cdot)\) \(\chi_{729}(334,\cdot)\) \(\chi_{729}(343,\cdot)\) \(\chi_{729}(361,\cdot)\) \(\chi_{729}(370,\cdot)\) \(\chi_{729}(388,\cdot)\) \(\chi_{729}(397,\cdot)\) ...

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 81 polynomial

Values on generators

\(2\) → \(e\left(\frac{23}{81}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 729 }(316, a) \) \(1\)\(1\)\(e\left(\frac{23}{81}\right)\)\(e\left(\frac{46}{81}\right)\)\(e\left(\frac{43}{81}\right)\)\(e\left(\frac{71}{81}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{29}{81}\right)\)\(e\left(\frac{22}{81}\right)\)\(e\left(\frac{13}{81}\right)\)\(e\left(\frac{11}{81}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 729 }(316,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 729 }(316,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 729 }(316,·),\chi_{ 729 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 729 }(316,·)) \;\) at \(\; a,b = \) e.g. 1,2