Student engagement has a trajectory (a timeline) that unfolds over time and can be shaped by different factors including learners' motivation, school conditions, and the nature of learning tasks. Such factors may result in either a stable, declining or fluctuating engagement trajectory. While research on online engagement is abundant, most authors have examined student engagement in a single course or two. Little research has been devoted to studying online longitudinal engagement, i.e., the evolution of student engagement over a full educational program. This learning analytics study examines the engagement states (sequences, successions, stability, and transitions) of 106 students in 1396 course enrollments over a full program. All data of students enrolled in the academic year 2014-2015, and their subsequent data in 2015-2016, 2016-2017, and 2017-2018 (15 courses) were collected. The engagement states were clustered using Hidden Markov Models (HMM) to uncover the hidden engagement trajectories which resulted in a mostly-engaged (33% of students), an intermediate (39.6%), and a troubled (27.4%) trajectory. The mostly-engaged trajectory was stable with infrequent changes, scored the highest, and was less likely to drop out. The troubled trajectory showed early disengagement, frequent dropouts and scored the lowest grades. The results of our study show how to identify early program disengagement (activities within the third decile) and when students may drop out (first year and early second year).