Don't know for sure what seeds you have called 'saffron' , because true saffron is derived from a crocus variety which is sterile and does not make seed. I suspect that they are a type of shallots which are sometimes called saffron. In that case treat them just as you would onion seed. Harvest when the tops dry in late summer or fall.

See how they develop, if they are some type of crocus, your Dixie cup is going to present a problem. Crocus are usually left in flat or pot the first year after germinating so the corm can begin to develop. After that they are moved up a pot size or carefully transplanted into the garden - and if a crocus, they would likely go late summer dormant too. See what type of top develops, maybe post a photo later, maybe someone can tell you if you have young onion type or crocus (which would mature much more slowly)

I only have a cell phone. This forum isn't mobile friendly for uploading pictures. The seeds where corn seed sized and white. The little sprouting has butterfly like wings leafs.

ArtichokeMe,
Go to Photobucket on your cell phone.
Download the app.
Take a pic of the plant and download the pic to Photobucket App on your cell phone.
After it's downloaded on photobucket, bring it up and get the HTML code, look under the pic and it will show you what it is.
You can put this right in a post on GW.
Try it.

Give it some time, newly germinated seedlings can be a little hard to identify, but I think most would describe crocus seedlings as looking more like grass....

Here is a link that might be useful: Saffron crocus seedlings/young plants

Funny story... I mislabeled my cup. It's a Florida giant watermelon.....