Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676000, base_ring=CyclotomicField(3900)) M = H._module chi = DirichletCharacter(H, M([0,975,546,1600]))
 
Copy content gp:[g,chi] = znchar(Mod(6009, 676000))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("676000.6009");
 

Basic properties

Modulus: \(676000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(338000\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3900\)
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 \(\chi_{338000}(259509,\cdot)\)
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 676000.bsu

\(\chi_{676000}(9,\cdot)\) \(\chi_{676000}(809,\cdot)\) \(\chi_{676000}(2089,\cdot)\) \(\chi_{676000}(2889,\cdot)\) \(\chi_{676000}(3129,\cdot)\) \(\chi_{676000}(3929,\cdot)\) \(\chi_{676000}(4169,\cdot)\) \(\chi_{676000}(4969,\cdot)\) \(\chi_{676000}(5209,\cdot)\) \(\chi_{676000}(6009,\cdot)\) \(\chi_{676000}(8329,\cdot)\) \(\chi_{676000}(9129,\cdot)\) \(\chi_{676000}(9369,\cdot)\) \(\chi_{676000}(10169,\cdot)\) \(\chi_{676000}(10409,\cdot)\) \(\chi_{676000}(11209,\cdot)\) \(\chi_{676000}(12489,\cdot)\) \(\chi_{676000}(13289,\cdot)\) \(\chi_{676000}(13529,\cdot)\) \(\chi_{676000}(14329,\cdot)\) \(\chi_{676000}(14569,\cdot)\) \(\chi_{676000}(15369,\cdot)\) \(\chi_{676000}(15609,\cdot)\) \(\chi_{676000}(16409,\cdot)\) \(\chi_{676000}(17689,\cdot)\) \(\chi_{676000}(18489,\cdot)\) \(\chi_{676000}(18729,\cdot)\) \(\chi_{676000}(19529,\cdot)\) \(\chi_{676000}(19769,\cdot)\) \(\chi_{676000}(20569,\cdot)\) ...

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: $\Q(\zeta_{3900})$
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 3900 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((126751,422501,389377,12001)\) → \((1,i,e\left(\frac{7}{50}\right),e\left(\frac{16}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 676000 }(6009, a) \) \(1\)\(1\)\(e\left(\frac{2347}{3900}\right)\)\(e\left(\frac{58}{195}\right)\)\(e\left(\frac{397}{1950}\right)\)\(e\left(\frac{571}{3900}\right)\)\(e\left(\frac{229}{1950}\right)\)\(e\left(\frac{281}{300}\right)\)\(e\left(\frac{1169}{1300}\right)\)\(e\left(\frac{13}{75}\right)\)\(e\left(\frac{1047}{1300}\right)\)\(e\left(\frac{3277}{3900}\right)\)
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)
 
\( \chi_{ 676000 }(6009,a) \;\) at \(\;a = \) e.g. 2