TITLE: Applications of full integral weight modular forms in elementary number theory
QUESTION [10 upvotes]: Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic functions as their Fourier coefficients. 
More to the point, my question is, apart from the relations you obtain between $\sigma_3, \sigma_5$ and $\sigma_7$ are there any applications of full integral weight modular forms (preferably cusp forms) to elementary number theory.

REPLY [2 votes]: Nobody seems to have mentioned  "the master"  of this subject, and his use of (classical) Eisenstein series to prove things like $p(5n+4) \equiv 0 \ ({\rm mod} \ 5)$ (here $p(m)$ is of course the usual partition function).
Here's  his proof  (prepared by Hardy), published in Math.Z (1921). B.Berndt published another   a bit shorter proof , which employs famous Ramanujan's differential equations. BTW, make sure you are familiar with the Ramanujan  "J-series" before you jump to (say) formula (2.2) in Berndt's paper :-)