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2018-01-15 18:06:19 +0100 commented answer Eigenvalues of hecke operators

@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.

2018-01-15 14:19:41 +0100 commented answer Eigenvalues of hecke operators

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!

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2018-01-15 14:00:44 +0100 commented answer Eigenvalues of hecke operators

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

2018-01-15 13:48:45 +0100 commented answer Eigenvalues of hecke operators

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?

2018-01-15 11:35:04 +0100 commented answer Eigenvalues of hecke operators

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?

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2018-01-15 11:21:23 +0100 commented answer Eigenvalues of hecke operators

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.

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2018-01-13 11:32:29 +0100 asked a question Eigenvalues of hecke operators

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.


2017-11-06 17:56:01 +0100 commented question How can I get back an expression for free variables in solve function.

Great, thank you for your help!

2017-11-06 16:36:26 +0100 commented question How can I get back an expression for free variables in solve function.

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

2017-11-06 14:38:21 +0100 commented question How can I get back an expression for free variables in solve function.

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 :)

2017-11-06 12:50:21 +0100 asked a question How can I get back an expression for free variables in solve function.

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?