Is there a list of the currently known $N$ such that $S_2(\Gamma_0(N)) = {0}$?
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"Currently known" is absurd; there is an explicit formula for $\operatorname{dim} S_2(N)$ and from this it is an elementary exercise to find all such $N$. There are exactly fifteen of them, namely $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25\}$.
answered May 1, 2014 at 15:41
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$\begingroup$ Yes, and more generally the formula for $S_k(\Gamma_1(N),\epsilon)$ where $\epsilon$ is a Nebentypus is also well-known, except for $k=1$. $\endgroup$– JoëlCommented May 1, 2014 at 17:36
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$\begingroup$ Does the OP assume that $N$ is an integer (rather than an ideal)? $\endgroup$– ConderCommented May 1, 2014 at 18:16