INSERT INTO profile user_id values (new.user_id);
Try putting () around the keys you are inserting.
INSERT INTO profile (user_id) values (new.user_id);
This is really an SQL question. Try running "sqlite" on the command line and get your SQL syntax right before jumping into Perl. It is easier to get error messages there too.
Are you posting in the right place? Check out Where do I post X? to know for sure.
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Want more info? How to link or
or How to display code and escape characters
are good places to start.