Here is the statement as I understand it to be, framed as a bijection of sets. My chief reference is the wonderful book Elliptic Curves, Modular Forms and their L-Functions by Álvaro Lozano-Robledo ...

Now that we have the new plugin mechanism, we can port the old examples back into the new system. Surjection Injection Bijection Note that we do not aim to perfectly replicate the old examples, but ...