Kusznetov trace formula deduction in Iwaniec-Kowalski

Question

I don't see how the authors deduce Theorem 16.9 of Iwaniec-Kowalski. They say in the line just before that they, after Cauchy-Schwarz, put (16.56) and (16.69) in (16.43) and it comes out, but I can't see it. After Cauchy-Schwarz we have as far as I can see terms like \[ \sum _\mathfrak a\int _{\pm \infty }\frac {|\tau (t,m)|^2dt}{t^3\cosh t}\] whereas they seem to have \[ \sum _\mathfrak a\int _{\pm \infty }\frac {e^{-(t/T)^2}|\tau (t,m)|^2dt}{\cosh t}.\] What am I seeing wrong, or what am I missing? I don't really understand the summation, so it's possible I'm missing something there, or I may be missing a bounding trick with $e^{-t^2}$.