Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(32)) M = H._module chi = DirichletCharacter(H, M([0,9,0]))
 
Copy content gp:[g,chi] = znchar(Mod(229, 384))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.229");
 

Basic properties

Modulus: \(384\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(128\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(32\)
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 \(\chi_{128}(101,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 384.v

\(\chi_{384}(13,\cdot)\) \(\chi_{384}(37,\cdot)\) \(\chi_{384}(61,\cdot)\) \(\chi_{384}(85,\cdot)\) \(\chi_{384}(109,\cdot)\) \(\chi_{384}(133,\cdot)\) \(\chi_{384}(157,\cdot)\) \(\chi_{384}(181,\cdot)\) \(\chi_{384}(205,\cdot)\) \(\chi_{384}(229,\cdot)\) \(\chi_{384}(253,\cdot)\) \(\chi_{384}(277,\cdot)\) \(\chi_{384}(301,\cdot)\) \(\chi_{384}(325,\cdot)\) \(\chi_{384}(349,\cdot)\) \(\chi_{384}(373,\cdot)\)

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: \(\Q(\zeta_{32})\)
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: \(\Q(\zeta_{128})^+\)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((127,133,257)\) → \((1,e\left(\frac{9}{32}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 384 }(229, a) \) \(1\)\(1\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{19}{32}\right)\)\(i\)
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)
 
\( \chi_{ 384 }(229,a) \;\) 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)
 
\( \tau_{ a }( \chi_{ 384 }(229,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(229,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(229,·)) \;\) at \(\; a,b = \) e.g. 1,2