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