How to win over my boyfriend's mother

Hey! So you need some background because there is a lot about this woman that she assumes about me that I would like to clear up. My boyfriend and I are very serious and I want a good relationship with her.
But she thinks I am a liar. I have been with my boyfriend for 3 years and I was raised to be black and white honest, even if it hurts other peoples' feelings. So I don't lie. She assumes that I sleep around with people even though I am fiercely loyal to her son and have been since day one. In fact, he was the one who was disfaithful to me for 6 months (when we first met so we are long past that but still— she has it wrong, obviously).
Her only "sources" are rumors she would hear from my old archery team we used to be on and once I quit, the rumors started again. Her son obviously has tried to change her mind but any time he defends me they fight. 
She has a family of 40 people who love me and they all try to help me but she seems like the only one who won't accept me. I'm a senior in high school, he is a sophomore in college (ages 18,19— will be ages 19,20 by early 2017). What can I do to show her that I'm a good person?