Let $E$ be an unknown elliptic curve over $\mathbb{Q}$.
Let $L(E, s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ be the L-function of $E$ and write $f(q) = \sum_{n=1}^{\infty} a_n q^n$.
I'm in a setting where I have found (as complex floats) a finite set $\{ q_1, q_2, \ldots, q_k \}$ of complex numbers in the open unit disc such that $f(q_i) = 0$, and this set is a complete set of representatives of the $\Gamma_0(N)$-orbits of such zeroes.
Can I now by some direct and explicit procedure compute the integers $a_2$, $a_3$, $a_4$, etc.?
(I also know $N$, so I can of course get a list of all elliptic curves with this conductor from LMFDB, and check numerically (for each of these curves) whether its associated modular form is zero on all of my values $q_i$. But this is not the kind of "direct and explicit" procedure I'm after.)
