> On S_2(Gamma_0(887)) the charpoly of T_2 has no root with +1
> eigenvalue, so the above computation does not suggest a bug in MAGMA
> in characteristic 2.

I have thought more and have decided that Mestre has made a slip and that
magma is probably OK here. What is going on at 2 in this 887 example?
Mestre says it's peu ramifiee and I haven't checked this but let's believe
it. He also says a_2=1. I bought this initially but I don't buy it now, I
made a slip when checking this. The disc at 2 is 2^6 and the factorization
is 1^4.1 so we are in Buhler's Case 17. The image of Decomp at 2 is then
A_4; fortunately A_4 in SL_2(F_4) is easy to understand, it seems to me
that it's just the upper triangular matrices. So mod 2 the repn is reducible
and so we must be ordinary. So a_2 is non-zero and is going to be the
eigenvalue of the unramified character in the bottom RH corner of the
local repn (see thms 2.5 and 2.6 or whatever, of Edixhoven's Inventiones 
weights paper). Indeed the image of inertia is C_2xC_2 i.e. 1 *;0 1. What
is left has order 3 and so in fact a_2 is an elt of GF(4) with order 3.
This checks out fine with everything and seems to indicate that whenever
Mestre says a_2=1 he means a_2=(non-triv cube root of 1 in GF(4)). Well
done for sorting this out!