An introduction that emphasises the historical development is Elliptic and Modular Functions from Gauss to Dedekind to Hecke by Ranjan Roy. The first chapter, The basic modular forms of the nineteenth century, describes much of the early history.

An introduction that emphasises the historical development is Elliptic and Modular Functions from Gauss to Dedekind to Hecke by Ranjan Roy. The first chapter, The basic modular forms of the nineteenth century, describes much of the early history.

The modular form of weight one in the OP was indeed published by Klein in 1878, but it appeared already in a 1858 paper by Hermite and was implicit in work by Gauss. (Roy writes extensively on Gauss' early contributions to modular forms in chapter 2.)