Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([32,27]))
 
Copy content gp:[g,chi] = znchar(Mod(49, 405))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.49");
 

Basic properties

Modulus: \(405\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(405\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(54\)
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 405.t

\(\chi_{405}(4,\cdot)\) \(\chi_{405}(34,\cdot)\) \(\chi_{405}(49,\cdot)\) \(\chi_{405}(79,\cdot)\) \(\chi_{405}(94,\cdot)\) \(\chi_{405}(124,\cdot)\) \(\chi_{405}(139,\cdot)\) \(\chi_{405}(169,\cdot)\) \(\chi_{405}(184,\cdot)\) \(\chi_{405}(214,\cdot)\) \(\chi_{405}(229,\cdot)\) \(\chi_{405}(259,\cdot)\) \(\chi_{405}(274,\cdot)\) \(\chi_{405}(304,\cdot)\) \(\chi_{405}(319,\cdot)\) \(\chi_{405}(349,\cdot)\) \(\chi_{405}(364,\cdot)\) \(\chi_{405}(394,\cdot)\)

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: \(\Q(\zeta_{27})\)
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 54 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((326,82)\) → \((e\left(\frac{16}{27}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 405 }(49, a) \) \(1\)\(1\)\(e\left(\frac{5}{54}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{53}{54}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{9}\right)\)
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)
 
\( \chi_{ 405 }(49,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 405 }(49,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 405 }(49,·),\chi_{ 405 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 405 }(49,·)) \;\) at \(\; a,b = \) e.g. 1,2