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TheBeiram gravatar image

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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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').

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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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'). We need the eigenvalues of the hecke operators since they correspond with the coefficients of the Fourier series.

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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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'). We need the eigenvalues of the hecke operators since they correspond with the coefficients of the Fourier series.

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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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'). We need the eigenvalues of the hecke operators since they correspond with the coefficients of the Fourier series.


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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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'). We need the eigenvalues of the hecke operators since they correspond with the coefficients of the Fourier series.


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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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. Say I have been trying found these matrices above, I can create k.< a >=NumberField(x^2+x-1), but then I don't know a represents the root of the polynomial. So the question for me is, how to proceed with this a. can I make Sage give me the eigenvalues of T3 and T5 in terms of 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'). We need the eigenvalues of the hecke operators since they correspond with the coefficients of the Fourier series.


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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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. 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 T3 and T5 in terms of a?

Furthermore, I am wondering how to get the jordan normal form, 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.


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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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. 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 T3 and T5 in terms of a?

Furthermore, I am wondering how to get the jordan normal form, form of T2, 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.


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updated 1 year ago

FrédéricC gravatar image

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 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=(21/2;21),T3=(31;43)T5=(21/2;21) so we have the following characteristic polynomials f2=x2+x1, f3=x25, f5=x2+2x4. 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. 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 T3 and T5 in terms of a?

Furthermore, I am wondering how to get the jordan normal form of T2, 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.