The government website has a checklist you can use. For people born between 1983 and 2000 this is the relevant page.
Based on it you’ll be a British citizen if when you were born at least one of your parents was either:
- a British citizen
- an Irish citizen living in the UK
- a citizen of a country that was in the EU or the EEA at the time AND living in the UK, and working or studying here
- a citizen of another country with one of the following immigration status:
- indefinite leave to remain (ILR)
- right of abode
- right of re-admission
As you mentioned your father had only a student visa when you were born, which is unfortunately not one of the categories mentioned above. You can check here on more details about these three categories.
Also he wasn't a citizen of an EU/EEA country at the time you were born as well.
Based on these details you are most likely not a British citizen.