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 ...
On the front page, the definition for bijection is given: A Bijection is an invertible function that converts back and forth between two different types, with the contract that a round-trip through ...
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