↑

Properties

Label 733.22
Modulus $733$
Conductor $733$
Order $61$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(733, base_ring=CyclotomicField(122)) M = H._module chi = DirichletCharacter(H, M([72]))
 
Copy content gp:[g,chi] = znchar(Mod(22, 733))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("733.22");
 

Basic properties

Modulus: \(733\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(733\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(61\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 733.g

\(\chi_{733}(10,\cdot)\) \(\chi_{733}(16,\cdot)\) \(\chi_{733}(22,\cdot)\) \(\chi_{733}(27,\cdot)\) \(\chi_{733}(58,\cdot)\) \(\chi_{733}(76,\cdot)\) \(\chi_{733}(93,\cdot)\) \(\chi_{733}(100,\cdot)\) \(\chi_{733}(134,\cdot)\) \(\chi_{733}(160,\cdot)\) \(\chi_{733}(183,\cdot)\) \(\chi_{733}(188,\cdot)\) \(\chi_{733}(195,\cdot)\) \(\chi_{733}(197,\cdot)\) \(\chi_{733}(206,\cdot)\) \(\chi_{733}(218,\cdot)\) \(\chi_{733}(220,\cdot)\) \(\chi_{733}(256,\cdot)\) \(\chi_{733}(263,\cdot)\) \(\chi_{733}(267,\cdot)\) \(\chi_{733}(270,\cdot)\) \(\chi_{733}(299,\cdot)\) \(\chi_{733}(312,\cdot)\) \(\chi_{733}(315,\cdot)\) \(\chi_{733}(333,\cdot)\) \(\chi_{733}(352,\cdot)\) \(\chi_{733}(361,\cdot)\) \(\chi_{733}(364,\cdot)\) \(\chi_{733}(386,\cdot)\) \(\chi_{733}(398,\cdot)\) ...

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: $\Q(\zeta_{61})$
Fixed field: Number field defined by a degree 61 polynomial

Values on generators

\(6\) → \(e\left(\frac{36}{61}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 733 }(22, a) \) \(1\)\(1\)\(e\left(\frac{27}{61}\right)\)\(e\left(\frac{9}{61}\right)\)\(e\left(\frac{54}{61}\right)\)\(e\left(\frac{5}{61}\right)\)\(e\left(\frac{36}{61}\right)\)\(e\left(\frac{37}{61}\right)\)\(e\left(\frac{20}{61}\right)\)\(e\left(\frac{18}{61}\right)\)\(e\left(\frac{32}{61}\right)\)\(e\left(\frac{31}{61}\right)\)
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)
 
\( \chi_{ 733 }(22,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 733 }(22,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 733 }(22,·),\chi_{ 733 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 733 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2