  We develop the theory and algorithms necessary to be able to verify the
strong Birch--Swinnerton-Dyer Conjecture for absolutely simple modular abelian
varieties over $\mathbf{Q}$. We apply our methods to all 28 Atkin--Lehner
quotients of $X_0(N)$ of genus $2$, all 97 genus $2$ curves from the LMFDB
whose Jacobian is of this type and six further curves originally found by Wang.
We are able to verify the strong BSD Conjecture unconditionally and exactly in
all these cases; this is the first time that strong BSD has been confirmed for
absolutely simple abelian varieties of dimension at least $2$. We also give an
example where we verify that the order of the Tate--Shafarevich group is $7^2$
and agrees with the order predicted by the BSD Conjecture.
