Hola Justin

It's difficult with these verbs that already contain this doubt nuance. I see sospechar and no sospechar like creer and no creer. When you use it in affirmative: Sospecho que... you need the indicative because you are declaring "what it is you suspect" - seen as a fact (or think, with Creo que...) 

When we make it negative: No sospecho que... (no creo que...) you are offering then some background information that is seen as "already known". It's a bit difficult though because using "no sospechar que" is not that common - you'd probably find that it is more natural to say "Estoy seguro de que ..."  (I'm certain that... ) and this always uses the indicative. 

Saludos