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Properties

Label 243.47
Modulus $243$
Conductor $243$
Order $162$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(162)) M = H._module chi = DirichletCharacter(H, M([61]))
 
Copy content gp:[g,chi] = znchar(Mod(47, 243))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("243.47");
 

Basic properties

Modulus: \(243\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(243\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(162\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 243.j

\(\chi_{243}(2,\cdot)\) \(\chi_{243}(5,\cdot)\) \(\chi_{243}(11,\cdot)\) \(\chi_{243}(14,\cdot)\) \(\chi_{243}(20,\cdot)\) \(\chi_{243}(23,\cdot)\) \(\chi_{243}(29,\cdot)\) \(\chi_{243}(32,\cdot)\) \(\chi_{243}(38,\cdot)\) \(\chi_{243}(41,\cdot)\) \(\chi_{243}(47,\cdot)\) \(\chi_{243}(50,\cdot)\) \(\chi_{243}(56,\cdot)\) \(\chi_{243}(59,\cdot)\) \(\chi_{243}(65,\cdot)\) \(\chi_{243}(68,\cdot)\) \(\chi_{243}(74,\cdot)\) \(\chi_{243}(77,\cdot)\) \(\chi_{243}(83,\cdot)\) \(\chi_{243}(86,\cdot)\) \(\chi_{243}(92,\cdot)\) \(\chi_{243}(95,\cdot)\) \(\chi_{243}(101,\cdot)\) \(\chi_{243}(104,\cdot)\) \(\chi_{243}(110,\cdot)\) \(\chi_{243}(113,\cdot)\) \(\chi_{243}(119,\cdot)\) \(\chi_{243}(122,\cdot)\) \(\chi_{243}(128,\cdot)\) \(\chi_{243}(131,\cdot)\) ...

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 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

\(2\) → \(e\left(\frac{61}{162}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 243 }(47, a) \) \(-1\)\(1\)\(e\left(\frac{61}{162}\right)\)\(e\left(\frac{61}{81}\right)\)\(e\left(\frac{107}{162}\right)\)\(e\left(\frac{29}{81}\right)\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{91}{162}\right)\)\(e\left(\frac{1}{81}\right)\)\(e\left(\frac{119}{162}\right)\)\(e\left(\frac{41}{81}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 243 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 243 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 243 }(47,·),\chi_{ 243 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 243 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2