Forget is a reporting verb.
Forget is a compound verb, formed with the preposition for and the verb get. Its inflection broadly follows get, and is therefore an irregular lexical verb; it has a third person form "forgets", the third person form "forgetting" (with a double T), an irregular preterite "forgot", and an irregular past participle "forgotten". The pattern follows the American inflection of get (get+got+gotten) rather than the British inflection (get+got+got).
"Forget+to-infinitive" is semantically different to "forget+gerund". For example, "I forgot to lock my house" means that "I didn't lock my house because of my bad memory" (CCQ: Did I lock my house? No.). Contrast that with "I forget locking my house" which means "I locked my house, but cannot recall the moment when I did so." (CCQ: Did I lock my house? Yes). This difference is also found in remember.
Instead of a gerund, it may be better to use a relative clause, e.g. "I forgot that I had locked my house".