What web-mail client are you using?
For Horde, I get:
Deprecated: Assigning the return value of new by reference is deprecated in /usr/local/cpanel/base/horde/lib/Horde/Perms/sql.php on line 83
Deprecated: Assigning the return value of new by reference is deprecated in /usr/local/cpanel/base/horde/lib/Horde/Perms/sql.php on line 116
Deprecated: Assigning the return value of new by reference is deprecated in /usr/local/cpanel/base/horde/lib/Horde/Perms/sql.php on line 153
Deprecated: Assigning the return value of new by reference is deprecated in /usr/local/cpanel/base/horde/lib/Horde/Prefs.php on line 864
Warning: Cannot modify header information - headers already sent by (output started at /usr/local/cpanel/base/horde/lib/Horde/Perms/sql.php:83) in/usr/local/cpanel/base/horde/index.php on line 97
For other clients, such as Roundcube and SquirrelMail, they work just fine.
Also, have you made sure that you're using the cPanel password for your account's webmail? If so, you should be able to log in just fine. (Or for that matter, each e-mail account's password...)
By the by, the errors I get for Horde is also on the absolut server.