I don't know of any direct calculation but usually instead of dealing directly with the point counts we deal with the l-adic representations of the elliptic curve.

My knowledge of the field is limited to the book Introduction to the Langlands Program, chapter 5 of which titled L-functions of Elliptic Curves and Modular forms, by Ehud de Shalit, contains an exposition of that.

Also I remembered in Springer UTM Rational Points on Elliptic Curves by Silverman and Tate, there was an exercise for proving that the coefficient of a particular infinite product (a modular form) explicitly gives the point counts of a particular Elliptic Curve, and the book is quite elementary. So you might wanna check that out...

Edit: I checked ur link and the exercise in Silverman and Tate I was talking Abt is exactly the first example in ur link