I too was fascinated by this quote. And then I came across a very interesting set of slides by none other than Prof. Ken Ribet (herein PKR) titled *The ﬁve fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms*

I'll try to summarize his slides below. As usual, any mistakes here are my own and not PKR's.

So, to start PKR bases his around discussing the topic in terms of counting, and solving problems equations.

This talk is about counting, and it’s about solving equations

In particular, Diophantine Equations.

PKR doesn't really discuss addition, subtraction, multiplication or division.
However he does provide simple answer to the question: *what the heck is modular form*:

Modular forms are special functions that are analogous to the
trigonometric functions like $\sin$, $\cos$, $\tan$,... in that they are
periodic in the same way that $\sin$ is periodic. (Recall the formula
$\sin(x + 2\pi) = \sin(x)$.) Modular forms have the periodicity of the
trigonometric functions plus enough extra symmetries that they
are essentially unchanged under a large group of substitutions.
Because of the symmetries, it is possible to write modular
forms as Fourier series $\sum_{m=0}^{\infty}{}$, where the $q$ here is a
shorthand for $e^{2\pi\iota\\z}$

Any how hope this helps