Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5175, base_ring=CyclotomicField(110)) M = H._module chi = DirichletCharacter(H, M([55,44,5]))
 
Copy content gp:[g,chi] = znchar(Mod(2006, 5175))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5175.2006");
 

Basic properties

Modulus: \(5175\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1725\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(110\)
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 \(\chi_{1725}(281,\cdot)\)
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 5175.cu

\(\chi_{5175}(296,\cdot)\) \(\chi_{5175}(341,\cdot)\) \(\chi_{5175}(431,\cdot)\) \(\chi_{5175}(521,\cdot)\) \(\chi_{5175}(566,\cdot)\) \(\chi_{5175}(746,\cdot)\) \(\chi_{5175}(881,\cdot)\) \(\chi_{5175}(971,\cdot)\) \(\chi_{5175}(1286,\cdot)\) \(\chi_{5175}(1331,\cdot)\) \(\chi_{5175}(1466,\cdot)\) \(\chi_{5175}(1556,\cdot)\) \(\chi_{5175}(1736,\cdot)\) \(\chi_{5175}(1781,\cdot)\) \(\chi_{5175}(1916,\cdot)\) \(\chi_{5175}(2006,\cdot)\) \(\chi_{5175}(2321,\cdot)\) \(\chi_{5175}(2366,\cdot)\) \(\chi_{5175}(2411,\cdot)\) \(\chi_{5175}(2591,\cdot)\) \(\chi_{5175}(2636,\cdot)\) \(\chi_{5175}(2771,\cdot)\) \(\chi_{5175}(2816,\cdot)\) \(\chi_{5175}(3041,\cdot)\) \(\chi_{5175}(3356,\cdot)\) \(\chi_{5175}(3446,\cdot)\) \(\chi_{5175}(3536,\cdot)\) \(\chi_{5175}(3671,\cdot)\) \(\chi_{5175}(3806,\cdot)\) \(\chi_{5175}(3986,\cdot)\) ...

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: $\Q(\zeta_{55})$
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 110 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((4601,3727,2926)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 5175 }(2006, a) \) \(1\)\(1\)\(e\left(\frac{109}{110}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{107}{110}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{47}{55}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{1}{55}\right)\)\(e\left(\frac{97}{110}\right)\)
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)
 
\( \chi_{ 5175 }(2006,a) \;\) at \(\;a = \) e.g. 2