Is it okay for me to send a picture of myself to a girl I like?

[David]

I go to college with this girl and I just met her this semester. We have three classes together, twice a week and sit next to each other and talk the entire time (starting 2 weeks ago as school has been going on for 4 weeks now) during two of them. We don't really text much though. I can't really tell if she likes me but she says random things to me during class and always makes gestures to get my attention. We relate pretty well with just about everything we talk about and I feel there is a good vibe. I really like her and she knows that because I let her listen to a song I wrote about her today. I didn't tell her it was for her but that she was the "inspiration" for it. She said she liked it but not in a really liking it kind of tone. I don't know if she has a boyfriend though but she has a picture of her with a guy as her Facebook profile picture so i'm going to go ahead and assume its a possibility.
I'm a really cutesy kinda guy