Where the slopes are (Q2764525)

Fix a prime number \(p\). Assume \(p\nmid N\), and let \(f\in S_k(N_p,\mathbb{C}_p):= S_k(N,\mathbb{Q})\otimes \mathbb{C}_p\) be an eigenform for the Atkin-Lehner \(U\) operator: \(U(f)=\lambda f\). Let \(\text{slope}(f)= \text{ord}_p(\lambda)\).NEWLINENEWLINE The author investigates numerically the distribution of the slopes of \(U\) for fixed level and varying weight. The computations suggests several interesting questions (conjectures): (i) the slopes are (much) smaller than expected (Questions 1, 2, 3, 4), (ii) the slopes are almost always integers (Question 5), (iii) one hopes for a representation-theoretic characterization of eigenforms that are of exceptional (see p. 85) or non-integral slope (Question 8).NEWLINENEWLINE In the case of classical modular forms, \textit{J.-P. Serre} [J. Am. Math. Soc. 10, 75--102 (1997; Zbl 0871.11032)] and \textit{J. B. Conrey}, \textit{W. Duke} and \textit{D. W. Farmer} [Acta Arith. 78, 405--409 (1997; Zbl 0876.11020)] investigated the distribution as \(k\to\infty\) of all the eigenvalues of the \(p\)th Hecke operator \(T_p\) corresponding to eigenforms of weight \(k\).