You could ask them to show you how they got to your $584 monthly.
The monthly payment is composed of 3 main parts
(1) Depreciation: this is the amount between the selling price and the residual divided by 36 (months). So your residual is $51,200 * 0.62 = $31,744. Based on your info, it seems like the sell price is $51,200 - $2000 - $500 - $1000 = $47,700 … ~$443/mo
(2) Interest: With the MF of 0.00038, that is < 1% interest so a couple of bucks a month
(3) Tax: 9-10% (WA?) * your monthly so maybe $40/mo
If you can figure out what is going on with each part broken down, it will be easy to see where you can get to $440. It seems like the best place for you is to figure out the actual discounts you are getting from MSRP - how much are they going to sell the car to you for? That will drive down (1) Depreciation … everything else is already good specially with the MF with your MSDs