Properties

Label 243.44
Modulus $243$
Conductor $81$
Order $54$
Real no
Primitive no
Minimal no
Parity odd

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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([41]))
 
pari: [g,chi] = znchar(Mod(44,243))
 

Basic properties

Modulus: \(243\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 243.h

\(\chi_{243}(8,\cdot)\) \(\chi_{243}(17,\cdot)\) \(\chi_{243}(35,\cdot)\) \(\chi_{243}(44,\cdot)\) \(\chi_{243}(62,\cdot)\) \(\chi_{243}(71,\cdot)\) \(\chi_{243}(89,\cdot)\) \(\chi_{243}(98,\cdot)\) \(\chi_{243}(116,\cdot)\) \(\chi_{243}(125,\cdot)\) \(\chi_{243}(143,\cdot)\) \(\chi_{243}(152,\cdot)\) \(\chi_{243}(170,\cdot)\) \(\chi_{243}(179,\cdot)\) \(\chi_{243}(197,\cdot)\) \(\chi_{243}(206,\cdot)\) \(\chi_{243}(224,\cdot)\) \(\chi_{243}(233,\cdot)\)

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 54 polynomial

Values on generators

\(2\) → \(e\left(\frac{41}{54}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 243 }(44, a) \) \(-1\)\(1\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{25}{54}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{47}{54}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{1}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 243 }(44,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 243 }(44,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 243 }(44,·),\chi_{ 243 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 243 }(44,·)) \;\) at \(\; a,b = \) e.g. 1,2