Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40310, base_ring=CyclotomicField(1932)) M = H._module chi = DirichletCharacter(H, M([483,207,518]))
 
Copy content gp:[g,chi] = znchar(Mod(617, 40310))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("40310.617");
 

Basic properties

Modulus: \(40310\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(20155\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1932\)
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 \(\chi_{20155}(617,\cdot)\)
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 40310.gc

\(\chi_{40310}(3,\cdot)\) \(\chi_{40310}(243,\cdot)\) \(\chi_{40310}(247,\cdot)\) \(\chi_{40310}(253,\cdot)\) \(\chi_{40310}(293,\cdot)\) \(\chi_{40310}(387,\cdot)\) \(\chi_{40310}(543,\cdot)\) \(\chi_{40310}(577,\cdot)\) \(\chi_{40310}(617,\cdot)\) \(\chi_{40310}(823,\cdot)\) \(\chi_{40310}(827,\cdot)\) \(\chi_{40310}(907,\cdot)\) \(\chi_{40310}(1063,\cdot)\) \(\chi_{40310}(1133,\cdot)\) \(\chi_{40310}(1197,\cdot)\) \(\chi_{40310}(1353,\cdot)\) \(\chi_{40310}(1407,\cdot)\) \(\chi_{40310}(1547,\cdot)\)

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: $\Q(\zeta_{1932})$
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 1932 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((24187,19461,16821)\) → \((i,e\left(\frac{3}{28}\right),e\left(\frac{37}{138}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 40310 }(617, a) \) \(-1\)\(1\)\(e\left(\frac{269}{966}\right)\)\(e\left(\frac{1819}{1932}\right)\)\(e\left(\frac{269}{483}\right)\)\(e\left(\frac{107}{1932}\right)\)\(e\left(\frac{1619}{1932}\right)\)\(e\left(\frac{13}{69}\right)\)\(e\left(\frac{1583}{1932}\right)\)\(e\left(\frac{425}{1932}\right)\)\(e\left(\frac{85}{644}\right)\)\(e\left(\frac{269}{322}\right)\)
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)
 
\( \chi_{ 40310 }(617,a) \;\) at \(\;a = \) e.g. 2