Hi,
I was wondering about good techniques that one can use to show that given certain coefficients, they are the Fourier coefficients of a cusp form, assuming we know the desired weight and level. I am aware of Weil's "converse theorem", but am not aware of any examples of it being used to prove something is a modular form. So, even a pointer to an example of that would be useful. I'm curious if there are other more direct methods as well. Thanks!



Verifying coefficients of modular forms
Maybe I can add something:


1. Wiles famous proof of Fermat's Last Theorem (and the later proof of modularity of all elliptic curves over $\mathbf{Q}$ by Breuil, Conrad, Diamond, Taylor) is a sort of ultimate example in which "given certain coefficients [associated to an elliptic curve by point count], they are [proved to be] the Fourier coefficients of a cusp form". This theorem also produces a general strategy for showing that a list of numbers are the coefficients of a cusp form in some cases -- relate those numbers to the point counts of an elliptic curve. Similar remarks apply for abelian varieties these days, due to Ribet's theorem that Serre's conjecture implies modularity of all abelian varieties over $\mathbf{Q}$ of ${\rm GL}\_2$-type.
2. There is something called the Faltings-Serre method, which is an alternative approach to proving modularity statements in particular cases. A google search for "Faltings Serre" will yield many hits. Here's a particular example: <http://www.math.wisc.edu/~boston/ell.pdf> . There is also a paper by Edray Goins that explains how to use the Faltings-Serre method to get new results in particular cases that go beyond (1), and it has a good list of references: <http://homepage.mac.com/ehgoins/papers/serre-faltings.pdf>
3. Another example is <http://wstein.org/papers/artin/>, which is a paper Kevin Buzzard and I wrote, in which we show that lists of numbers (traces of Frobenius) associated to certain A\_5 Galois representations are the coefficients of weight 1 modular forms. One can get the same result by "pure thought" these days as an application of Khare-Wintenberger's proof of Serre's conjecture. But still, there is some value in the computational techniques introduced in that paper. There are similar ideas in Joe Buhler's Ph.D. thesis and also in Springer Lecture Notes in Math Volume 1585.
4. There is a Theorem of Sturm (see, e.g., Theorem 9.18 of my modular forms book <http://wstein.org/books/modform/stein-modform.pdf> and the references) that tends to be **very** relevant to your question in practice. Sturm's result basically tells you that if the first "few" coefficients of two $q$-expansions of modular forms agree modulo a prime, then all terms agree.


This list could go on and on. Also, there are similar ideas and techniques for Hilbert modular forms. 


