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$R$ 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 whose which is the splitting field is 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.

$R$ 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 whose splitting field is $x^4 + 3x^2 - 7x + 4$. 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
(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.
$R$ 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 whose splitting field is $x^4 + 3x^2 - 7x + 4$. 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. Or you can write such programs yourself! [which is what I did and which is why magma can do it ;-) ]. More self-promotion available (including the algorithm) at