Extreme values of $L$-functions of newforms

In 2008, Soundararajan showed that there exists a normalized Hecke eigenform $f$ of weight $k$ and level one such that $$L(1\mathrm{/}2,f)\ge \exp ((1+o(1))\sqrt{\frac{2\log k}{\log \log k}})$$ for sufficiently large $k\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$. In this note, we show that for any $\epsilon >0$ and for all sufficiently large $k\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, the number of normalized Hecke eigenforms of weight $k$ and level one for which $$L(1\mathrm{/}2,f)\ge \exp \left(1.41\sqrt{\frac{\log k}{\log \log k}}\right)$$ is ${\gg }_{\epsilon }{k}^{1-\epsilon }$. For an odd fundamental discriminant $D$, let $B_k(\mathrm{\mid }D\mathrm{\mid })$ be the set of all cuspidal normalized Hecke eigenforms of weight $k$ and level dividing $\mathrm{\mid }D\mathrm{\mid }$. When the real primitive Dirichlet character ${\chi }_D$ satisfies ${\chi }_D(-1)=i^k$, we investigate the number of $f\in B_k(\mathrm{\mid }D\mathrm{\mid })$ for which $L(1\mathrm{/}2,f\otimes {\chi }_D)$ takes extremal values.