Let \(H_k\) be the set of normalized weight-\(k\) Hecke eigenforms for \(\text{SL}_2(\mathbb{Z})\), and for \(f\in H_k\) let \(N(\alpha, T, \text{sym}^m f)\) be the zero counting function for the \(L\)-series of \(\text{sym}^m f\). Then for \(1\leq m\leq 4\) and any \(\varepsilon>0\) it is shown that \[ \sum_{f\in H_k} N(\alpha,T,\text{sym}^mf)\ll_\varepsilon T^2 k^{29(1- \alpha)+\varepsilon}. \] In fact slightly sharper bounds are proven.) Thus for “almost all” \(f\in H_k\) one has \(L(s,\text{sym}^m f)\neq 0\) for \(\sigma\geq {31\over 32}\) and \(|t|\leq k^{1/32}\). For such \(f\), and for \(m\leq 4\), it is shown further that \[ L(1, \text{sym}^m f)\ll (\log\log k)^{A_m},\quad L(1, \text{sym}^m f)= \Omega((\log\log k)^{A_m}) \] and \[ L(1, \text{sym}^m f)^{-1}\ll (\log\log k)^{B_m},\quad L(1, \text{sym}^m f)^{-1}= \Omega((\log\log k)^{B_m}) \] for explicitly given exponents \(A_m\), \(B_m\). The implied constants are investigated, in analogy with the well known results for \(\zeta(1+ it)\).
Finally, it is shown that there are forms \(f_+\), \(f_-\), \(f_0\in H_k\) for which there are many primes \(p\leq(\log k)^A\) for which the Fourier coefficients \(\lambda_f(p)\) are nearly 2; or nearly \(-2\); or nearly 0, respectively.
The proofs use standard techniques extended to automorphic symmetric power \(L\)-functions, together with a large sieve inequality for the coefficients \(\lambda_{\text{sym}^m f}(l)\) as \(l\) varies.