The first odd degree-2 Artin representation for which the Artin conjecture was proved At the DeKalb conference on Hilbert's problems, John Tate gave a masterly survey of Problem 9, the General Reciprocity Law.  He ends with a discussion of the Langlands Programme, especially the case of odd Artin representations $R$ of degree $2$.  Let me quote from the final paragraph of the written version of his talk :

Another reason for the relationship's eluding Artin and Hecke may be the fact that explicit non-dihedral numerical examples are hard to find.  Indeed at the time of the DeKalb conference, none was known !  I concluded the oral presentation of the paper there by explaining that, in the hope of finding one, I had looked for non-dihedral $R$'s of low conductor, $N$, and had found an $R$ with $N=133=7\cdot19$  which I hoped might be amenable to computation.  After the talk, Atkin suggested that the labor involved might be considerably reduced by systematic use of $w_7$ and $w_{19}$.  Armed with his theory of the $w$'s, four Harvard students, D. Flath, R. Kottwitz, J. Tunnell, J. Weisinger and I succeeded in the next month in proving (by relatively easy hand computation) the existence of the corresponding new form $f_R$ of weight $1$ and level $133$ predicted by Langlands.

My first question is : what was $R$ ?  The second question is :  how would you verify today (on a computer) the existence of $f_R$, without invoking any of the theorems which have been proved in the meantime ?  
 A: $R$ is a 2-dimensional conductor 133 representation of the absolute Galois group of the rationals into $GL(2,\mathbf{C})$, whose associated representation to $PGL(2,\mathbf{C})$ cuts out the $A_4$ extension of the rationals which is the splitting field of $x^4 + 3x^2 - 7x + 4$. To verify the existence of the weight 1 form of level 133 you can just fire up a magma session (magma is a computer algebra package) and ask it to compute the weight 1 level 133 forms, and the dihedral weight 1 level 133 forms, and then note that there are more weight 1 level 133 forms than dihedral ones. So it's pretty easy now. Or you can write such programs yourself! [which is what I did and which is why magma can do it ;-) ]. More shameless self-promotion available (including the algorithm) at
http://www2.imperial.ac.uk/~buzzard/maths/research/papers/wt1.pdf
(where I e.g. explain that there's even a level 124 non-dihedral form, and give a description of the $S_4$ form of smallest conductor). However the ideas all first appeared in print in Joe Buhler's thesis donkey's years ago, where he finds an $A_5$ form: Springer LNM654.
