I am beginning to learn SAGE. I have been working on a project that requires me to have an (orthogonal) eigenbasis for $S_2^{\text{new}}(\Gamma_0(N))$. Is there an easy way to get that?
I think I can use CuspForms(Gamma0(N),2).new_submodule().hecke_matrix(p) CuspForms(Gamma0(N),2).new_submodule().hecke_matrix(p) (for many primes) with some linear algebra to, after some effort, build an eigenbasis. But since this is something basic in the theory of modular forms, I thought it would be already implemented in SAGE (but I could not find it). Is that so?
edit:----
Thank you very much for the answers below, I am about to accept them. Though, I have been trying to use those routines and encountered some issues. In some cases, they don't seem to give me all newforms. I am trying to understand why. For example, for N=89, k=2
sage: S_new = CuspForms(89,2).new_submodule()
sage: S_new.dimension()
7
sage: Newforms(89,names='a')
[q - q^2 - q^3 - q^4 - q^5 + O(q^6),
q + q^2 + 2*q^3 - q^4 - 2*q^5 + O(q^6),
q + a2*q^2 + (-1/2*a2^4 + 1/2*a2^3 + 7/2*a2^2 - 5/2*a2 - 4)*q^3 + (a2^2 - 2)*q^4 + (-a2^2 + 4)*q^5 + O(q^6)]
Why does it not calculate the remaining eigenforms in this case? Is it because the coefficient fields are too complicated? LMFDB says one of them has degree 140 https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/?level=89&weight=2&search_type=List
Using Cremona's answer with (...).q_eigenform(50,names='a') d.q_eigenform(50,names='a') yields the same.
I see that I can change the base field to $\bar{\mathbb{Q}}$ to get the full collection:
sage: Newforms(89,names='a',base_ring=QQbar)
[q - 2.47725271?*q^2 - 2.759353854?*q^3 + 4.1367810?*q^4 - 2.13678098?*q^5 + O(q^6),
q - 2.098127745?*q^2 + 2.3452649700?*q^3 + 2.402140034?*q^4 - 0.402140035?*q^5 + O(q^6),
q - q^2 - q^3 - q^4 - q^5 + O(q^6),
q - 0.74595374?*q^2 - 0.54990973?*q^3 - 1.44355303?*q^4 + 3.44355303?*q^5 + O(q^6),
q + 1.00000000000000?*q^2 + 2.00000000000000?*q^3 - 1.00000000000000?*q^4 - 2.00000000000000?*q^5 + O(q^6),
q + 1.62790662528?*q^2 - 0.14890720385?*q^3 + 0.6500799807?*q^4 + 1.34992001936?*q^5 + O(q^6),
q + 2.693427562?*q^2 - 1.8870941866?*q^3 + 5.254552031?*q^4 - 3.254552031?*q^5 + O(q^6)]
But then I have to work with these approximation and lose some information. For example, I cannot recover the Hecke eigenvalue fields from these forms.
I think one solution would be to compute the eigenvalues of the Hecke matrixes (say, for the rational basis that SAGE gives me via CuspForms(89,2).basis()). CuspForms(89,2).basis()). And then try to recover the coefficient fields from them. To then finally use Newforms(89,names='a',base_ring=K), Newforms(89,names='a',base_ring=K) with K being the compositum of coefficient fields. The problem is that it takes too long in cases like this one.
I am wondering if that is all that I can do in this case? Or maybe there is a better way to use those commands, that to avoids these issues?
