You can find a detailed treatment for all cuspidal representations of $GL(2)$ over a number field in Péter Maga's [thesis][1], see his Proposition 3.2 on Page 20. Note that this is really what you need, because the residue appearing in the proposition equals, up to an explicit constant depending on the level and the number field (which is essentially the volume of the fundamental domain), the ratio of the squared norm of a newvector and the squared norm of the corresponding vector in the Whittaker model, cf. (3.3) and (3.4) in the mentioned work. This is the adelic analogue of (0.5) in Hoffstein-Lockhart who normalize $f$ to have $\|f\|=1$.

**Added.** I add more details responding to the OP's comment. There is no need to be familiar with the general theory to use the above results. Let 
$$f(z)=\sum_{n=1}^\infty a(n)e(nz)$$ be a holomorphic newform of weight $k$ and level $N$, so that $a(n)=a(1)\lambda(n)n^{\frac{k-1}{2}}$, where $\lambda(n)$ is the $n$-th Hecke eigenvalue normalized so that the Ramanujan conjecture (proved by Deligne for this setting) reads $|\lambda(p)|\leq 2$ for $p$ prime. Using the Rankin-Selberg unfolding technique (see e.g. Section 1.6 in Bump: Automorphic forms and representations) we readily get
$$\frac{3}{\pi}\int_{\Gamma_0(N)\backslash\mathcal{H}} y^k|f(x+iy)|^2 \frac{dxdy}{y^2}=(4\pi)^{-k}\Gamma(k)\cdot\mathrm{res}_{s=1}\sum_{n=1}^\infty\frac{|a(n)|^2}{n^{s+k-1}},$$
that is,
$$\mathrm{res}_{s=1}\sum_{n=1}^\infty\frac{|\lambda(n)|^2}{n^{s}}=\frac{3}{\pi}\cdot\frac{(4\pi)^k}{\Gamma(k)}\cdot\frac{\|f\|^2}{|a(1)|^2}.$$
The Dirichlet series on the left hand side agrees with $L(s,\pi\otimes\tilde\pi)$ apart from having different Euler factors at the ramified primes $p\mid N$, which are easy to estimate. Here I denoted by $\pi$ the cuspidal representation generated by $f(z)$. Hence the general bound mentioned in my original post above yields
$$ (Nk)^{-\epsilon}\ll \frac{(4\pi)^k}{\Gamma(k)}\cdot\frac{\|f\|^2}{|a(1)|^2}\ll (Nk)^\epsilon.$$
Of course this is a standard result that can be used in a research paper without any further comment. If $f(z)$ is a non CM form, the bounds can be improved further, see e.g. the Appendix by Goldfeld-Hoffstein-Lieman to the paper by Hoffstein-Lockhart.

  [1]: http://www.renyi.hu/~magap/publications/dissertations/phd.pdf