Applications of full integral weight modular forms in elementary number theory 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.
 A: Personally, I regard the Dirichlet coefficients of any automorphic $L$-function arithmetic. A nice application of the Fourier coefficients of full level Maass cusp forms is Motohashi's improvement for the error term in the binary additive divisor problem and the asymptotic formula for the fourth moment of the Riemann zeta function. See in particular his papers here and Theorem 5.2 in his book.
A: Another little thing: the simplest Siegel-Weil formulas, equating holomorphic Eisenstein series and linear combinations of theta series attached to positive-definite quadratic forms, can be arranged to be about level-one or small-level things.
Edit: and add Klingen's proof of rationality properties of special values of zeta functions of totally real number fields.
A: 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 :-)
A: Perhaps this is "out of bounds" given the phrasing of the question, but those Eisenstein series you mention don't just have divisor sums as coefficients - the constant term is a special value of the Riemann zeta function.
This implies all sorts of neat stuff.  The various relations between divisor sums that you mention come with relations between zeta values.  These give very nice congruences, in particular.  
You can take this to reasoning pretty far to deduce things like $p$-adic interpolation of zeta values a la Kubota-Leopoldt from the much simpler interpolation properties of the divisor sum functions.  This was done by Serre in the 1973 paper ("Formes modulaires et fontiones zeta $p$-adiques") that gave birth to the theory of $p$-adic modular forms.
