A symmetric $p$-adic symbol for triples of modular forms

In 2014, Darmon and Rotger defined the Garrett-Rankin triple product $p$-adic $L$- function and related it to the image of certain diagonal cycles under the $p$-adic Abel- Jacobi map. We introduce a new $p$-adic triple symbol based on this $p$-adic $L$- function and show that it satisfies symmetry relations, when permuting the three input modular forms. We also provide computational examples illustrating this symmetry property. To do so, we extend Lauder's algorithm to allow for ordinary projections of nearly overconvergent modular forms -- not just overconvergent modular forms -- as well as certain projections over spaces of non-zero slope. Our work also gives an efficient method to calculate certain Poincar\'e pairings in higher weight, which may be of independent interest.