TITLE: How is representation theory used in modular/automorphic forms?
QUESTION [48 upvotes]: 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?

REPLY [2 votes]: Observation:
The modular forms/automorphic representations should be seen as generalization of Hecke quasi characters, which are simply characters $\chi: K^{\times }\backslash \mathbb{A}_K^{\times} \rightarrow \mathbb{C}^{\times}$. Tate's thesis gives a proof of the functional equation for the associated L-functions purely in terms of representation theory. He studies the right regular representation on the space of some functions $f:K^{\times }\backslash \mathbb{A} \rightarrow \mathbb{C}$. This is the so called $\mathrm{GL}_1$ case, since $\mathrm{GL}_1(K)=K^{\times}$. 
The next obvious choice is then to consider other reductive groups instead of $\mathrm{GL}_1$ e.g. $\mathrm{GL}_2$.
One remark how I think about Tate's thesis: The right regular representation on an (locally compact) abelian groups is in direct connection with its Fourier transform. The functional equation can be seen as an adelic version of the Poisson summation formula.
Observation:
The special linear group $\mathrm{SL}_2(\mathbb{R})$ acts on upper half plane $\mathbb{H}$ by Moebius transformations. Moreover this group is actually the group of all biholomorphic mappings $\mathbb{H} \rightarrow \mathbb{H}$. The isotropy group of $\mathrm{i}$ is $\mathrm{SO}_2$, i.e. the group of elements, which fix $\mathrm{i}$. Since the action is associative, i.e. for any $x, y \in \mathbb{H}$ there exists $g \in \mathrm{SL}_2(\mathbb{R})$ such that $gx = y$, we get an isomorphism of the orbit space with the space we act on. Hence, we actually have $\mathbb{H} \cong \mathrm{SL}_2(\mathbb{R}) /\mathrm{SO}_2$.
Observation
You can lift through weak approximation Dirichlet character $\chi : (\mathbb{Z}/n \mathbb{Z})^{\times} \rightarrow \mathbb{C}^{\times}$ to the adele space $K^{\times }\backslash \mathbb{A}_K^{\times}$. 
You can proceed similiar with strong approximation and obtain that the double coset space
$ SL_2(\mathbb{Q}) \backslash SL_2( \mathbb{A} ) / K $ is actually isomorphic to the orbit space $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}.$ Here $K$ means the product of all $\mathrm{SL}_2(\mathbb{Z}_p)$ for the finite places and $\mathrm{SO}_2$ for the archimedian place.