To express degrees of necessity we can use various modal verbs.
must is used when the necessity is dictated by the speaker's authority. have [got] to is used when the necessity is dictated by an external force or event, which the speaker cannot control.
(doctor to patient)
(patient to themself)
In all these examples there is an optional element, which defeats somewhat the element of necessity, i.e. we needn't stay late but we can if we want to.
These show that calling your mother is the right thing to do. There is no real difference between ought to and should, but ought to is perhaps a little stronger.
We use needn't to show an action is optional - I can do it if I want to. We use mustn't to say an action is forbidden - I have no choice.