What is the ruling on my work as technical support for the bank's programs and products directed at companies, knowing that my mother swore she would not eat from my money if I worked at the bank?

Means are subject to the rulings of their objectives. Your work in programming for the bank supports the promotion of its usurious services. Since whoever assists in a sin is a partner in it, your work is considered an assistance to the bank in its usurious transactions, due to the Almighty’s saying: "And do not cooperate in sin and aggression." The Hadith that cursed the one who consumes usury, the one who feeds it, its scribe, and its two witnesses, indicates the prohibition of assisting in falsehood. Therefore, you must leave this job, even if the other job pays less, for Allah blesses the lawful and obliterates the unlawful. Your family must enjoin you to do good and forbid you from evil.