Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(684)) M = H._module chi = DirichletCharacter(H, M([171,10]))
 
Copy content gp:[g,chi] = znchar(Mod(32, 9025))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9025.32");
 

Basic properties

Modulus: \(9025\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1805\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(684\)
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 \(\chi_{1805}(32,\cdot)\)
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 9025.ck

\(\chi_{9025}(32,\cdot)\) \(\chi_{9025}(143,\cdot)\) \(\chi_{9025}(193,\cdot)\) \(\chi_{9025}(243,\cdot)\) \(\chi_{9025}(257,\cdot)\) \(\chi_{9025}(268,\cdot)\) \(\chi_{9025}(318,\cdot)\) \(\chi_{9025}(357,\cdot)\) \(\chi_{9025}(382,\cdot)\) \(\chi_{9025}(393,\cdot)\) \(\chi_{9025}(432,\cdot)\) \(\chi_{9025}(507,\cdot)\) \(\chi_{9025}(618,\cdot)\) \(\chi_{9025}(718,\cdot)\) \(\chi_{9025}(732,\cdot)\) \(\chi_{9025}(743,\cdot)\) \(\chi_{9025}(782,\cdot)\) \(\chi_{9025}(793,\cdot)\) \(\chi_{9025}(832,\cdot)\) \(\chi_{9025}(857,\cdot)\) \(\chi_{9025}(868,\cdot)\) \(\chi_{9025}(907,\cdot)\) \(\chi_{9025}(982,\cdot)\) \(\chi_{9025}(1093,\cdot)\) \(\chi_{9025}(1143,\cdot)\) \(\chi_{9025}(1193,\cdot)\) \(\chi_{9025}(1207,\cdot)\) \(\chi_{9025}(1218,\cdot)\) \(\chi_{9025}(1257,\cdot)\) \(\chi_{9025}(1268,\cdot)\) ...

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: $\Q(\zeta_{684})$
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 684 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((5777,3251)\) → \((i,e\left(\frac{5}{342}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 9025 }(32, a) \) \(1\)\(1\)\(e\left(\frac{181}{684}\right)\)\(e\left(\frac{535}{684}\right)\)\(e\left(\frac{181}{342}\right)\)\(e\left(\frac{8}{171}\right)\)\(e\left(\frac{101}{228}\right)\)\(e\left(\frac{181}{228}\right)\)\(e\left(\frac{193}{342}\right)\)\(e\left(\frac{28}{57}\right)\)\(e\left(\frac{71}{228}\right)\)\(e\left(\frac{383}{684}\right)\)
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)
 
\( \chi_{ 9025 }(32,a) \;\) at \(\;a = \) e.g. 2