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In Schulze-Pillot, Yenrice "Petersson Products of bases of spaces of cusp forms" Theorem 11, they mention a bound which might answer b). For a newform $f$ of level $N$ it states $$\langle f,f\rangle\geq \bigg(4\pi \mathrm{e}^{4\pi} N\cdot \prod_{p \mid N}\left(1+\frac{1}{p}\right)\bigg)^{-1}\,.$$ This is achieved by the estimation $$\int_{\mathcal F} \left|f(z)\right|^2 y^{k-2}\,\mathrm{d}x\,\mathrm{d}y\geq \int_1^\infty \exp \left(-4\pi y\right)\,\mathrm{d}y\,,$$ where $\mathcal F$ is the usual fundamental domain. Consider that $\left[\Gamma_1:\Gamma_0(N)\right]=N\cdot \prod_{p \mid N}\left(1+\frac{1}{p}\right)$ is the normalizing factor for the Petersson inner product.