Properties

Label 243.37
Modulus $243$
Conductor $81$
Order $27$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([32]))
 
pari: [g,chi] = znchar(Mod(37,243))
 

Basic properties

Modulus: \(243\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(27\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 243.g

\(\chi_{243}(10,\cdot)\) \(\chi_{243}(19,\cdot)\) \(\chi_{243}(37,\cdot)\) \(\chi_{243}(46,\cdot)\) \(\chi_{243}(64,\cdot)\) \(\chi_{243}(73,\cdot)\) \(\chi_{243}(91,\cdot)\) \(\chi_{243}(100,\cdot)\) \(\chi_{243}(118,\cdot)\) \(\chi_{243}(127,\cdot)\) \(\chi_{243}(145,\cdot)\) \(\chi_{243}(154,\cdot)\) \(\chi_{243}(172,\cdot)\) \(\chi_{243}(181,\cdot)\) \(\chi_{243}(199,\cdot)\) \(\chi_{243}(208,\cdot)\) \(\chi_{243}(226,\cdot)\) \(\chi_{243}(235,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 27 polynomial

Values on generators

\(2\) → \(e\left(\frac{16}{27}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 243 }(37, a) \) \(1\)\(1\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{10}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 243 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 243 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 243 }(37,·),\chi_{ 243 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 243 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2