Hecke Bound for Cusp - Modular Forms

[binbagsss]

Homework Statement

i have a questions on the piece of lecture notes attached:

2. Homework Equations

The Attempt at a Solution

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I agree 2) of proposition 2.12 holds once we have 1). I thought I understood the general idea of 1), however, my reasoning would hold for $M_k$ it does not depend on $f(t)$ being a cusp and so it must be wrong. This was what I thought was happening:

$q=e^{2\pi i n (u+iv)} ~ e^{-v}$ for large v, and exponential dominates over $v ^ {x}$ ( v>0 as on upper plane )

This would ofc still hold if I included some constant term, I would still get the same quantity is bounded.
can someone please tel me where I have gone wrong with the above reasoning?

 

[binbagsss]

$q=e^{2\pi i n (u+iv)} \approx e^{-v}$ for large v, and exponential dominates over $v ^ {x}$ ( v>0 as on upper plane )

edited apologies latex error