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!
Maybe I can add something:
This list could go on and on. Also, there are similar ideas and techniques for Hilbert modular forms. 


It depends on what you mean by "certain coefficients" and how they're given. This sort of sounds like you have a (finite) handful of coefficients in a $q$expansion and want to try to "fill them out" to a modular form of prescribed weight and level. If this is the case, it is pretty easy. This is because the space of such forms is finitedimensional, and a basis of the space is readily computed (using, for example, the SAGE software). This reduces the problem to basic linear algebra. If, on the other hand, you have an entire $q$expansion, it's not so clear. You can truncate and employ the above procedure. I'll just mention three things: 1) If your expansion isn't a modular form, then this will eventually (by taking larger chunks) tell you so. 2) If it is a modular form (but you have no a priori knowledge of this), then it just gives you a lot of evidence that your expansion is a modular form, but doesn't prove it. Of course, if you know more about your coefficients you might be able to go back and prove that it is the form evidenced above. 3) If you do happen to know that your expansion is a modular form of given weight and level and just want to know "what it is," then this method will eventually provably determine what it is (in terms of the basis) by taking large enough chunks. As for the converse theorem... you'd certainly need all the coefficients, but I have a hard time imagining that method to be very practical (but I'm not very analytically minded). 


I think Ramsey's answer hits the important points, but I wanted to comment on the converse theorem. The converse theorem takes as an input a Dirichlet series (more generally, an $L$function) whose analytic continuation (plus quite a bit more) is already known and tells you that it actually came from an automorphic form. Unfortunately, the only methods that I know for proving analytic continuation require knowing that it already comes from an automorphic form. So it certainly sounds circular, and I've only seen it applied to proving global functoriality (for quasisplit classical groups, small symmetric powers, etc). Let me quickly(!) explain the argument: Modularity / potential automorphy methods (e.g. Serre's Conjecture, Modularity Theorem) exist, too. In the weight 2 case, given enough information about the coefficients and a sufficient amount of pluck, it is not inconceivable that you could show that the coefficients actually come from an elliptic curve (or several elliptic curves if your modular form is not an eigenform), then use modularity to deduce that they come from a modular form. 


If you assume that the modular form is a newform with prescribed weight and level (EDIT : you also need to know the eigenvalue of the AtkinLehner involution), then its Fourier coefficients can sometimes be guessed using the fact that the associated Lseries satisfies a functional equation. The point is that using the functional equation, the Lseries can be computed by a rapidly convergent series, so it is only necessary to have few Fourier coefficients at hand to compute an approximation of it. This even gives a very pratical and powerful method for finding modular forms, at least when the level is small. You should look at the following example written by Tim Dokchitser : http://magma.maths.usyd.edu.au/magma/handbook/text/1392#15270 Dokchitser's algorithm for computing Lfunctions has been implemented in Sage and Magma. Note that the method outlined here is (to my knowledge) purely experimental, in the sense that it doesn't prove that the resulting qexpansion is modular. 


( tried posting this earlier today, apologies if it appears twice) For an example of a direct application of Weil's converse theorem, see Shimura's original formulation of the Shimura lift (Annals 1973). He constructs a map from modular forms of halfintegral weight to modular forms of integral weight by defining an Lseries built out of the Fourier coefficients of the input form, and spends most of the paper proving the analytic properties needed for the converse theorem. 

