...
  INSERT INTO profile user_id values (new.user_id);
  ...
[download]

Try putting () around the keys you are inserting.

  ...
  INSERT INTO profile (user_id) values (new.user_id);
  ...
[download]

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.