This question is based off these notes by Preston Wake about Iwasawa invariants and Hida Families. In the notes, the author asks "why" the elliptic curve $11A3$ has $\mu$-invariant equal to $2$ for the prime $p=5$. He offers a heuristic justification by saying that for all primes $\ell \neq 11$, we have the congruence

$$a_{\ell} \equiv \chi(\ell) + \ell \chi^{-1}(\ell) \mod 5^2$$

where $\chi: (\mathbf{Z}/11\mathbf{Z})^{\times} \to (\mathbf{Z}/11\mathbf{Z})^{\times}$ is the character that sends $2$ to $6$. Here, as usual, $a_{\ell}$ is defined by $a_{\ell} = |E(\mathbf{F}_{\ell})| - (\ell+1)$. According to the notes, the first term $\chi(\ell)$ in the above sum "comes from Eisensteiness" and the second term $\ell \chi^{-1}(\ell)$ in the sum "comes from extra reducibility".

My question is: where does the equation above come from? And what does it have to do with the $\mu$-invariant of $11A3$? Because a priori, it is simply a congruence between the various $a_{\ell}$'s; it seems unrelated to $\mu$-invariants. More generally, I'd like to ask: what is the concrete link between the $\mu$-invariants of elliptic curves, and congruences between Fourier coefficients of modular forms like the one shown above? Is there a reference that explains this connection?

To be clear, I've read about how Hida families of modular forms can shed light on the $\mu$-invariants of modular forms. (As an example, in the paper of Pollack, Wake, and Weston.) My question, however, is about how Hida families of modular forms give us info about the $\mu$-invariants of elliptic curves. The equation above, for example, seems to suggest that we can gleam information about the $\mu$-invariants of elliptic curves by examining congruences between their Fourier coefficients. I'm not too clear on the link between those two sides of the story.

So if anyone knew of a reference specifically about congruences between modular forms and $\mu$-invariants of elliptic curves, I'd be immensely grateful. Thanks for the help!