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Properties

Label 729.685
Modulus $729$
Conductor $243$
Order $81$
Real no
Primitive no
Minimal no
Parity even

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(729, base_ring=CyclotomicField(162)) M = H._module chi = DirichletCharacter(H, M([14]))
 
Copy content gp:[g,chi] = znchar(Mod(685, 729))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("729.685");
 

Basic properties

Modulus: \(729\)
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: \(81\)
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: no, induced from \(\chi_{243}(103,\cdot)\)
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: no
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 729.i

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

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: $\Q(\zeta_{81})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 81 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 729 }(685, a) \) \(1\)\(1\)\(e\left(\frac{7}{81}\right)\)\(e\left(\frac{14}{81}\right)\)\(e\left(\frac{80}{81}\right)\)\(e\left(\frac{4}{81}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{37}{81}\right)\)\(e\left(\frac{56}{81}\right)\)\(e\left(\frac{11}{81}\right)\)\(e\left(\frac{28}{81}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 729 }(685,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 729 }(685,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 729 }(685,·),\chi_{ 729 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 729 }(685,·)) \;\) at \(\; a,b = \) e.g. 1,2