depending on your school, you get a new spell at level 35 and 55, assuming you have completed certain grizzleheim and wintertusk quests. you dont go to your teacher to get them. other npc's in hte spiral give you a quset or two for these spells.
Morgrim Darksword Magus (level 30) Necromancer
P.S. when do i get to see the king? i heard a rumor that you have to fight him. is the rumor true?