need some help with my first cover letter

Ok... I need some help with my cover letter. I am a sophmore, non-target school, ~3.2 GPA and my dream would be to break into trading. I know I need some kind of experience this summer so I was going to try and get a part-time, unpaid internship in PWM in my city. This cover letter is to the branch manager of the Chattanooga, TN pwm branch for UBS. This is the first cover letter I have ever written, but Im gonna write many more and send them out this week so I need to see if I'm on the right track. This is going to be a challenge because I have no experience-related or otherwise-that I can tie into my qualifications. Just to clarify, this is obviously not for an open position, I'm just sending these to Branch managers in hopes that someone will meet with me.

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