It is well-known that, given a normalized eigenform $f=\sum a\_n q^n$, its coefficients $a\_n$ generate a number field $K\_f$. 


In their 1995 [paper](http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf) "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the time of writing, very little was known about what sort of number fields could arise as some $K\_f$. They do claim, however, that $K\_f$ must be totally real or CM. This claim is made just before Lemma 1.37, on page 40 of the copy I linked to.


This is probably standard knowledge among experts, but I'm having trouble finding a reference, so my questions are:


1) Can someone please provide a reference for this claim?


2) Is this still the state of the art, or do we now know more about what types of fields can appear as $K\_f$ for some $f$? What if we restrict our attention to weight $k=2$?


Thank you!



> 
> Edit: In my question, I originally just wrote "modular form" instead of "normalized eigenform". Thanks to @Stopple for pointing this out! Also, I originally claimed the paper was published in 2007, but Kevin Buzzard pointed out it was published in 1995. Thanks Kevin!
> 
> 
> 



Number Fields Arising from Newforms
For (1), see Ribet's wonderful article *Galois representations attached to eigenforms with Nebentypus* (<http://dx.doi.org/10.1007/BFb0063943>). It's proposition 3.2.


