In Schulze-Pillot, Yenrice "Petersson Products of bases of spaces of cusp forms" Theorem 11, they mention a bound with might answer b). For a newform $f$ of level $N$ it states $\langle f,f\rangle\geq(4\pi e^{4pi} N\cdot \prod_{p|N}(1+\frac{1}{p}))^{-1}$.
Nodt Greenish
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