Properties

Label 243.103
Modulus $243$
Conductor $243$
Order $81$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(243, base_ring=CyclotomicField(162))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14]))
 
pari: [g,chi] = znchar(Mod(103,243))
 

Basic properties

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

Galois orbit 243.i

\(\chi_{243}(4,\cdot)\) \(\chi_{243}(7,\cdot)\) \(\chi_{243}(13,\cdot)\) \(\chi_{243}(16,\cdot)\) \(\chi_{243}(22,\cdot)\) \(\chi_{243}(25,\cdot)\) \(\chi_{243}(31,\cdot)\) \(\chi_{243}(34,\cdot)\) \(\chi_{243}(40,\cdot)\) \(\chi_{243}(43,\cdot)\) \(\chi_{243}(49,\cdot)\) \(\chi_{243}(52,\cdot)\) \(\chi_{243}(58,\cdot)\) \(\chi_{243}(61,\cdot)\) \(\chi_{243}(67,\cdot)\) \(\chi_{243}(70,\cdot)\) \(\chi_{243}(76,\cdot)\) \(\chi_{243}(79,\cdot)\) \(\chi_{243}(85,\cdot)\) \(\chi_{243}(88,\cdot)\) \(\chi_{243}(94,\cdot)\) \(\chi_{243}(97,\cdot)\) \(\chi_{243}(103,\cdot)\) \(\chi_{243}(106,\cdot)\) \(\chi_{243}(112,\cdot)\) \(\chi_{243}(115,\cdot)\) \(\chi_{243}(121,\cdot)\) \(\chi_{243}(124,\cdot)\) \(\chi_{243}(130,\cdot)\) \(\chi_{243}(133,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\(2\) → \(e\left(\frac{7}{81}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 243 }(103, a) \) \(1\)\(1\)\(e\left(\frac{7}{81}\right)\)\(e\left(\frac{14}{81}\right)\)\(e\left(\frac{80}{81}\right)\)\(e\left(\frac{4}{81}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{37}{81}\right)\)\(e\left(\frac{56}{81}\right)\)\(e\left(\frac{11}{81}\right)\)\(e\left(\frac{28}{81}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 243 }(103,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 243 }(103,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 243 }(103,·),\chi_{ 243 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 243 }(103,·)) \;\) at \(\; a,b = \) e.g. 1,2