Suppose that \(K\) is a Galois number field over \(\mathbb{Q}\). Let \(B\) be a non-CM abelian variety over \(K\) satisfying the following two conditions:
1) \(B\) is a variety over \(\mathbb{Q}\) admitting isogenies \(\mu_{\sigma} : {^\sigma}B \to B\) compatible with \(\mathrm{End}(B)\) defined over \(K\) for all \(\sigma\) in the Galois group \(\mathrm{Gal}(K:Q)\) and
2) \(\mathrm{End}_K(B)\) is a division algebra with center being a number field \(E\) such that the index \(t \leq 2\), and the reduced degree \(t[E:\mathbb{Q}] = \mathrm{dim}(B)\). These abelian varieties are called \(K\)-building blocks.
An abelian variety \(B\) over \(K\) is called strongly modular if its \(L\)-function \(L(B/K;s)\) is equivalent to a product of \(L\)-series of newforms over \(\mathbb{Q}\) for congruence subgroups of the form \(\Gamma_1(N)\). The main result of this paper is a characterization of strongly modular abelian varieties \(B\) over \(K\) using \(K\)-building blocks. More precisely, if \(B\) is a \(K\)-simple abelian variety, then \(B\) is strongly modular over \(K\) if and only if \(B\) is a \(K\)-building block that satisfies two extra conditions: 1) \(K\) over \(\mathbb{Q}\) is abelian;
2) the cocycle class \([c_{B/K}]\) is in the subgroup \(\mathrm{Ext}(G,E^*) \subset H^2(G,E^*)\) consisting of symmetric cocycle classes. In the last section, the authors also provide some concrete examples of abelian surfaces with quaternionic multiplication as applications of the general results.