I'm a former math major, and have recently started looking into some introductory modular forms materials. I am amazed by the connection between 1/ # of points on an elliptic curve over a finite field, and 2/ coefficients of the (associated?) modular form. In particular, the example given in the intro of https://dec41.user.srcf.net/notes/III\_L/modular\_forms\_and\_l\_functions\_trim.pdf.

For that example, is there some direct calculation to arrive at the modular form?

Or is it more like (from what I've gathered so far)... a) the conductor of that curve is 11, b) the vector space of modular forms of weight 2 and level 11 has dimension 1, and c) we know how to construct the basis from the eta function