Welcome, and congrats on your new car! Unfortunately there's no easy or cheap workaround to a lost key. The key, immobilizer, and ECU all communicate and share codes and if the codes don't agree, the ECU won't activate the injectors.
The easiest route is probably getting a new key, and a good mobile locksmith can cut it and program it. Replacement keys are specific to model years, with 00-01 needing an "R" key and 02-05 an "N" key. The RF identifier with the code is manufactured into the key, not the fob.
If you can do your own wrenching, you can remove your ECU and ship it to a tuner who knows how to "remove the immobilizer". This deletes the code-specific check and any key that mechanically fits into your ignition will now start your car.
If you really do your own wrenching, you can sometimes find a used set of a key, ignition cylinder, immobilizer, and ECU and swap all those components. The set has to match your year, engine, trim, and transmission type. Unless it comes with door locks and a trunk lock, you still will need a key made for those. Sometimes tuners will offer an ECU for sale with the code check already removed. These still have to match your car, and you'd need to find and cut a key. Most Eclipse tuners are private enthusiasts/hobbyists, not businesses.
Someone recently went through this and posted a nice writeup. I think it was
@3g moneypit.