For a project I am trying to build a program that computes a basis for cusp forms of weight 2 over Γ0(N). At one point, I want to represent eigenvalues of multiple Hecke operators. For example, (for N=23) I have found the following matrices for the Hecke operators: T2=(−2−1/2;21),T3=(31;−4−3)T5=(−2−1/2;21) so we have the following characteristic polynomials f2=x2+x−1, f3=x2−5, f5=x2+2x−4. Suppose we denote the solution of f2 (eigenvalues of T2) by a and it's conjugate. Then we can denote the solution of f3 by 2a+1 and of f5 by 2a. I am trying to figure out how I can 'make' Sage represent these eigenvalues in this way. I have been trying k.< a >=NumberField(x^2+x-1), but then I don't know how to proceed with this a.
Remark: A little context, the goal for this project is to represent newforms in a similar way as Sage does, when you ask for example Newforms(23,2,names='a').