TheBeiram's profile - activity

@dan_fulea, I have one more issue actually, the above method doesn't seem to work for all matrices. For example, the matrix $T_2$ for $N=41$ is $\begin{pmatrix} -1 & -1 & 0 \ 0 & 2 & 2 \ 2& 1/2& -2 \end{pmatrix}$, and if I take a number field over it's minimum polynomial, and I change the basis of the matrix (exactly as earlier) I get an error when I ask for the eigenvalues : 'need a real or complex embedding to convert a non rational element of a number field into an algebraic number'. I am afraid there is a deeper mathematical problem here, but maybe you know what is happening here.

Ah I didn't see your comment before. But still, change_ring() doesn't always seems to work. But the other solution is also ok. Thanks a lot!

Ok, I guess I solved it by just saying matrix(K,2,2,[T2[i] for i in srange(2)])

Yes this is what I need. Only thing is that my matrices come out of some algorithm, so I would like to transform them into matrices over $K$, instead of construct them as above. So given some matrix, how do I change the ring. change_ring doesn't seem to do the trick. I could just put the entries into matrix() one by one again, but is there a more direct way?

And what I am also wondering, if I have any matrix, (for instance T2) could I make the Jordan Form in terms of 'a'. You did this here, but it only works because you are working with CuspForms. But could we do this more generally?

Thank you for the long answer! I am afraid it does not completely solve my problem, but it is still interesting. The thing is that my matrices don't come from Sage, but from my own code that computes a basis of the homology and it's hecke operators (which is the purpose of my project), which corresponds to a basis of the cusp forms. So I am trying to make something independent of the command CuspForms. The slightly cryptical sentence I have edited now, hopefully it makes more sense.

For a project I am trying to build a program that computes a basis for cusp forms of weight 2 over $\Gamma_0(N)$. 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: $ T_2=\begin{pmatrix} -2 & -1/2 ; 2 & 1 \end{pmatrix}, T_3=\begin{pmatrix} 3 & 1 ;
-4 & -3 \end{pmatrix} \quad T_5=\begin{pmatrix} -2 & -1/2 ; 2 & 1 \end{pmatrix} \quad $ so we have the following characteristic polynomials $f_2=x^2+x-1$, $f_3=x^2-5$, $f_5=x^2 +2x-4$. Suppose we denote the solution of $f_2$ (eigenvalues of $T_2$) by $a$ and it's conjugate. Then we can denote the solution of $f_3$ by $2a+1$ and of $f_5$ by $2a$. I am trying to figure out how I can 'make' Sage represent these eigenvalues in this way. Say I have found these matrices above, I can create k.< a >=NumberField(x^2+x-1), then $a$ represents the root of the polynomial. So the question for me is, how can I make Sage give me the eigenvalues of $T_3$ and $T_5$ in terms of $a$?

Furthermore, I am wondering how to get the jordan normal form of $T_2$, or eigenmatrix in terms of $a$. If I ask for jordan_form, I get an error, and if I ask for eigenmatrix_right, I only get a numerical expression.

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'). We need the eigenvalues of the hecke operators since they correspond with the coefficients of the Fourier series.

Great, thank you for your help!

Yes that sounds nice. Any idea how to get rid of the many zeros?

Yes, I can see that. And I would just like to have one of these solutions, but probably this is not possible. The reason I am looking for this is that I have a C-vectorspace defined over some symbols (in this case, just a random example, the symbols A,B,C,D,E,F,G) and I want to quotient this space by a space spanned by some linear combinations of these symbols (in this case A+B==0, etc). When I did this by hand, I used the solved function, and could then see that a basis for my quotient space would be <a,-g,d>. To implement this process, I thought of also using the solve function, but it needs to give me an expression for the free variables. But probably this is not really the right approach :)

I have a number of linear equations on some symbols A,B,C,D,E,F,G. When I solve them in Sagemath, I get some free variables in the solution.

$A,B,C,D,E,F,G=var('A,B,C,D,E,F,G')$

$eqns=[A+B==0,C+D+E==0,F+G==0,A+E-F==0]$

$solution=solve(eqns,A,B,C,D,E,F,G)$

Sage gives the following solution: $[[A == r1, B == -r1, C == r1 - r2 - r3, D == r3, E == -r1 + r2, F == r2, G == -r2]]$

Now I can ask Sage to give me an expression of any combination of the symbols, for example: $(A+C).subs(solution)$, then I get $2*r1 - r2 - r3$. For my purpose I would now like to have an expression for any of the free variables. If I assign $r1=var('r1')$ and ask Sage $s1.subs(solution)$ I get back $r1$ again. But I would lik to get back an expression in terms of $A,B,C,D,E,F,G$. Any suggestions on how to do this?