Periods of fibre products of elliptic surfaces and the Gamma conjecture

We provide an algorithm for computing a basis of homology of fibre products of elliptic surfaces over ${\mathbb{P}}^1$, along with the corresponding intersection product and period matrices. We use this data to investigate the Gamma conjecture for Calabi-Yau threefolds obtained in this manner. We find a formula that works for all operators of a list of 105 fibre products, as well as for fourth order operators of the Calabi-Yau database. This algorithm comes with a SageMath implementation.