Exponential sum associated to Maass cups forms of level $N$

We consider the $L$-function associated with a nonzero Maass cusp form $f$ of weight $0$, level $N$, and Laplace eigenvalue $1/4+r^2$. Let $t(n)$ be the normalized Fourier coefficient corresponding to the Maass cusp form $f$. I need the estimate of $$\sum_{n\le T}t(n)e^{2\pi i n x},$$ where $x \in \mathbb{R}$ and $T$ is a large number.

From Lemma $5.3$ of the book by H. Iwaniec 1, we have \begin{equation} \sum_{n\le T}\lambda(n)e^{2\pi i n x}\ll T^{1/2}\log(2T), \quad \quad (1) \end{equation} where $\lambda(n)$ is the normalized Fourier coefficient of holomorphic cusp forms of weight $k$ and level $N$.

My question: is eq. $(1)$ true for Maass cusp forms of level $N$?

From Theorem 3 of Hafner 2, we know that eq. $(1)$ holds for Maass cusp forms of level $1$. Can anyone let me know how eq. $(1)$ can be extended to Maass cusp forms of level $N$?

Also, from Theorem 8.1 (see page 111) of 3, can we say that eq. (1) is true for Maass cusp forms of any level $N$?