Got a new T-mobile GS3 phone and I put a Solavei pre-paid card in it but...

the internet is insufferably slow. The facebook app won't even open so it's obviously a big problem, maybe there's some weird option I forgot to access or something as it should use T-mobile's bands.

Do you or did you change any of the APN's?

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That sounds highly complicated but will give it a try.

I tried to do it but where it says " send a picture to yourself". I tried to sent the picture but it has a loading circle, normal text keep sending normaly so the data is the problem.

Well your device is not on their approved list, but the bands for a t-mo phone are right. Talk to their customer service

But is the T-mobile branded GS3, I'll talk to them thanks a lot.

Edit: The fact that I could add a new APN means the phone is unlocked? because I got an update and it came from T-mobile.

I have the same phone, the SGH-T999, T-mobile branded. It worked fine for a month but I ran into problems with the seller on eBay (doesn't appear that it is a truly unlocked phone, they are also not the authorized user so T-mobile can't send them the "SIM Subsidiary unlock code"; I'm working with T-mobile now, faxed them my receipt and they said they would send the unlock code within 10-14 days).

When it was working this phone screamed on the network! I'm in the process of switching over from Sprint (was never able to get 4G at home with Sprint; I get 4G in the garage with Solavei) and really want to get this phone working again.

From what I know it's not a fully unlocked phone unless you purchase from a authorized seller. You need to talk with T-Mobile to get it unlocked. Usually users actually in a contract with T-Mobile can ask for the Unlock Code if they stay in good standing with the carrier.