Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups

We exhibit examples of geometrically simple abelian surfaces $A\mathrm{/}\mathbb{Q}$ with conductor bounded by $(10000{)}^2$ whose Tate–Shafarevich groups contain a subgroup isomorphic to $(\mathbb{Z}\mathrm{/}p\mathbb{Z}{)}^2$ for each $p=5,7,11,13$. To find these examples we generalise work of Cremona–Freitas to enumerate all congruences of a certain type between pairs of weight $2$ newforms $f\in S_2^{\mathrm{n}\mathrm{e}\mathrm{w}}({\mathrm{\Gamma }}_0(N))$ and $g\in S_2^{\mathrm{n}\mathrm{e}\mathrm{w}}({\mathrm{\Gamma }}_0(M))$ contained in the LMFDB (i.e., with $N,M<10000$) and with coefficient fields of degree $\le 4$. Passing from the modular forms to the corresponding abelian varieties we use visibility to (unconditionally) prove the existence of non-trivial elements of the Tate–Shafarevich group. Finally we construct an example of an abelian surface with $(\mathbb{Z}\mathrm{/}7\mathbb{Z}{)}^2\subset \mathrm{S}\mathrm{h}\mathrm{a}(A\mathrm{/}\mathbb{Q})$ which is (conjecturally) not visible in any abelian threefold.