Be creative here there are no wrong ideas.
I would like the Spartan laser to be a weapon personally, just called something different "Charge Rifle" was a name I had in mind
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Not really as a generic new weapon type, but for new exotics: Handcannon that goes into special slot. Terrible damage on shielded enemies, but tripled precision damage on unshielded ones and is insanely quick to ready like the plan c High RoF auto rifle that goes into special slot. Meager damage on unshielded enemies, but will melt shields away with range and accuracy rivaling that of snipers, along with a contained "suppressed" effect that prevents the target from regaining shields. Lasts a short while, but stacks a couple times the longer you sustain fire on an unshielded enemy. Both make these former primaries fit into the special scenarios archetype that special weapons are supposed to have