What are the applications of modular forms in number theory? I am new to the topic, so I'm trying to get an overview. I am aware of the relation between modular forms and $L$-series (but don't know what that does) and FLT.
Are there other applications of modular forms other than counting problems (by obtaining the coefficients of a series) in number theory?
A short list would be sufficient but a little more detail with that would be helpful.
EDIT:
I am aware of this post but my question is specifically on number theory.
 A: *

*Fibonacci and
Lucas Perfect Powers
Modular forms are used to demonstrate that the only perfect powers in
the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect
powers in the Lucas sequence are 1 and 4.


*The
Lebesgue-Nagell equation
Modular forms are used to solve the Lebesgue-Nagell equation.


*Densest sphere packing
Modular forms find the lattice with the densest sphere packing problem in dimensions 8 ($E_8$ lattice) and in dimension 24 (Leech lattice).


*Ramanujan's constant
Modular forms explain why $e^{\pi\sqrt{163}}$ is so close to an integer.
A: Modular forms are used to solve Fermat's last theorem, Mock modular forms are used in black holes theory. I read also in Quanta magazine that Eisenstein series are used to compute what we call Monster group.
https://d2r55xnwy6nx47.cloudfront.net/uploads/2017/08/symmetry-algebra-and-the-monster-20170817.pdf
Here is an interesting article to see the beautiful application of modular forms in astronomy https://m.facebook.com/story.php?story_fbid=795545447309716&id=247304225467177
