Translation of "falsely" into Latin

For it was falsely rumored that Caesar was approaching, and that his cavalry were already at the gates.

Caesar enim adventare iam iamque et adesse eius equites falso nuntiabantur.

Rhescuporis was removed to Alexandria, and there attempting or falsely charged with attempting escape, was put to death.

Rhescuporis Alexandriam devectus atque illic fugam temptans an ficto crimine interficitur.