Your mom must have a really good reason as to why she doesn't want you to live with your dad. You don't have to necessarily live with your dad; you can always go visit him once in a while. Personally I think that at 13 a girl should stay with her mom so I think it's for the best.
She loves you to much to let you go. Tell her you want to live with your dad for change and if you love me you should let me. Tell her you love her but you miss your dad, maybe you can do like every other week seeing the different parents.
Well, I've been in your situation and I ended up staying with my mom. Looking back on it, i should've gone with my dad but thats because my mom and I are 2 different people. You're old enough by law (if you live in Ca) to make that decision yourself so if you want to live with your dad, you should talk to him and see if that's okay with him and just politely let your mom know. If there are problems with your mom than maybe you should explain to her that it might be best for your relationship if you go with your dad. Hope this helps.