Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(690, base_ring=CyclotomicField(44)) M = H._module chi = DirichletCharacter(H, M([22,33,42]))
 
Copy content gp:[g,chi] = znchar(Mod(83, 690))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("690.83");
 

Basic properties

Modulus: \(690\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(345\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(44\)
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 \(\chi_{345}(83,\cdot)\)
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: 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 690.u

\(\chi_{690}(17,\cdot)\) \(\chi_{690}(53,\cdot)\) \(\chi_{690}(83,\cdot)\) \(\chi_{690}(107,\cdot)\) \(\chi_{690}(113,\cdot)\) \(\chi_{690}(143,\cdot)\) \(\chi_{690}(203,\cdot)\) \(\chi_{690}(227,\cdot)\) \(\chi_{690}(263,\cdot)\) \(\chi_{690}(287,\cdot)\) \(\chi_{690}(293,\cdot)\) \(\chi_{690}(383,\cdot)\) \(\chi_{690}(467,\cdot)\) \(\chi_{690}(497,\cdot)\) \(\chi_{690}(503,\cdot)\) \(\chi_{690}(527,\cdot)\) \(\chi_{690}(557,\cdot)\) \(\chi_{690}(563,\cdot)\) \(\chi_{690}(617,\cdot)\) \(\chi_{690}(677,\cdot)\)

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: \(\Q(\zeta_{44})\)
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 44 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((461,277,511)\) → \((-1,-i,e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 690 }(83, a) \) \(-1\)\(1\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{1}{44}\right)\)
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)
 
\( \chi_{ 690 }(83,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 690 }(83,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 690 }(83,·),\chi_{ 690 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 690 }(83,·)) \;\) at \(\; a,b = \) e.g. 1,2