I asked this on mathstackexchange, but got no answer.

Let $N\geq1$ be an integer, and let $\mathbb{T}$ be the Hecke algebra acting on the cusp forms of weight k and level $\Gamma_0(N)$.

Then $\mathbb{T}$ is finite and flat over $\mathbb{Z}$, but the proof I know for this fact is slightly roundabout, by letting it act on homology.

How can one compute $\mathbb{T}$ in explicit examples? By which I mean, writing a presentation for it. I am not necessarily looking for an algorithm, but all the examples I have seen are pretty degenerate (i.e. level 1, weight 12 or level 23, weight 2), and I would like to get some better feel for it. Moreover, it seems to me this sort of data cannot be easily found on the LMFDB.

Answers to slight variations of this question are also very welcome (i.e. level $\Gamma_1(N)$, or all forms instead of just cusp forms).

Thanks!