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.
Categories: math.NT