Hello and thank you for choosing just answer,

The only thing I can think of is that they didn't set the timing right, Very common mistake to happen with technicians but this is just my theory, It could be that something else was not done right or it might be another fault with the vehicle which I doubt.

I think the timing theory matches your description perfectly since it will cause a lack of performance and a slightly rough idle and the check engine light will come on.

I highly suggest you do not drive your vehicle until you contact the dealership, Tell them about what's going on and have them decide if it's OK to drive it to them or tow it to them, If they say it's OK to drive it and then something goes south they can't blame it on you driving the vehicle since they said it's OK to do so! Also that dealership better fix your car right since every dealership job comes with a year warranty and 12,000 miles. If they claim it's something else then have them show you and fully explain what's going on, Usually dealerships are pretty honest when they make a mistake and they will go above and beyond to make it right but in no way should you feel like they are going to put this on you actually you should be upset that they didn't fix it right the first time and have them figure out what's going on.

Just as a reminder if your Mazda dealership is giving you trouble concerning a recent repair that went south the first step to do is to talk to their service manager or service director. If they are not helping then contact Mazda's customer service no. at 1-800-866-1998.

Please remember to call the dealership prior to driving your vehicle so they would become liable in case something goes south with the problem, If they say no tow it to us then make it clear to them that they are paying for this tow since the vehicle is in the condition it's in because of their mistake and not yours. They should take care of tow expense automatically but it doesn't hurt to make sure of this.

If you have any other questions please let me know, If you are satisfied with my answer please rate it so I can get compensated for my time. if you are not satisfied please do not rate my answer and simply replay back and I'll do my best to help more. Thank you.