How can i trust my girlfriend again,?

Those emails are wrong of her because she didn't need to ask for her ex's number if she's already in a relationship with someone she loves. But, remember that this was a year ago. And maybe she's changed since then. She claims that's as far as it went. Maybe it is, maybe it isn't. But, you don't know. And you can't go assuming things in your head if you have no proof. Has she ever lied or deceived you before? If not then just let this go. For her to be really apologetic and remorseful shows how guilty she feels. This will be enough to set her straight and realize she can never mess up like this. Gain your trust back by having her delete all guys off her phone, email, messenger, etc. Ones who she had romantic feelings for such as ex's (not friends). Because she wouldn't need them if she's happily in a long term relationship. And if she doesn't have any then just let it go and tell her she has one more chance and you are forgiving her. Don't annoy her and hold this over her head because if you continue to rant about how much you don't trust her when she actually IS faithful, it will make her actually want to do what your accusing her of. Let this go and try to heal.