Extreme values of LL‐functions of newforms

In 2008, Soundararajan showed that there exists a normalized Hecke eigenform f of weight k and level one such that L (1/2, f) ~~ ( (1 + o (1) ) 2 k{ k }) for sufficiently large k 0 4. In this note, we show that for any >0 and for all sufficiently large k 0 4, the number of normalized Hecke eigenforms of weight k and level one for which L (1/2, f) ~~ (1. 41 k { k }) is _ k^1-. For an odd fundamental discriminant D, let B₊ (|D|) be the set of all cuspidal normalized Hecke eigenforms of weight k and level dividing |D|. When the real primitive Dirichlet character D satisfies D (-1) = iᵏ, we investigate the number of f B₊ (|D|) for which L (1/2, f D) takes extremal values.