There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and representation theory?

I have a basic grounding in the complex analytic theory of modular forms (their dimension formulas, how they classify isomorphism classes of elliptic curves, some basic examples of level N modular forms and their relation to torsion points on elliptic curves, series expansions, theta functions, Hecke operators). This is all with an undergraduate background in complex analysis and algebra (Galois theory). I also know a little bit about the basics of algebraic number theory and algebraic geometry, if that helps. More importantly, I have a basic background in the representation theory of finite groups. My question is, then, could one example how modular forms and/or theta functions relate to representations of groups?

I'm asking this in part because I imagine a number of students with similar background as I have would have learned about modular forms and thus might be interested to understand how they relate to representation theory, despite not having an extensive background in more advanced results in algebraic geometry and commutative algebra needed for advanced study in the field.

Here are some ideas which might bear fruit: In analytic number theory, one often sees sums over characters - but characters are also very relevant in representation theory. In particular, Jacobi sums come up in both number theory and representation theory (and quadratic forms then relate to theta functions). Is there a connection here?

In addition, Hecke operators are symmetric-like sums over elements of groups, which would suggest a strong connection to representation theory.

Or is the connection to representations of $\mathrm{SL}_2(\mathbb{Z})$? Quotients of this group appear as Galois groups of extensions of spaces of modular forms, so they might be given representations by acting on these spaces?

The point of listing ideas is to show the kind of intuition I might be looking for. One of my ideas might be fruitful, or they all might have nothing to do with why representation theory connects to modular functions. The point is that I'm looking for basic ideas that someone with an elementary background might be able to understand.

I also added "reference request" because I imagine there might be a text which is at my level and discusses these ideas.

EDIT: The answer of paul garett here actually gives a nice history of how modular forms came to be viewed in terms of representations theory: What is the difference between an automorphic form and a modular form?

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