It depends on what you mean by "certain coefficients" and how their 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 finite-dimensional, 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).

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 finite-dimensional, 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).