Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL\_2(Z)$ and let $\lambda\_f(n)$ be the $n$-th normalized Fourier coefficient of $f.$ Can someone provide me with an effective estimate for this infinite product
 $$\prod\_{\lambda\_f(p)=0} \left(1-\frac{1}{p+1}\right).$$


Thanks in advance.



Effective estimate for this infinite product over Hecke eigenvalues
In general, this product is hard to work with, and good upper/lower bounds require a bit of work. I'll also assume the simplest case of $k=12$, in which case $\lambda\_f(n)$ is the famous Ramanujan tau function.


Since the density of primes $p$ for which $\lambda\_f(p)=0$ is zero (a result first proved by Serre his paper mentioned above), the product converges absolutely. We expect that $\lambda\_f(p)\neq0$ for all $p$ (Lehmer's conjecture), so this product should simply be 1.


For upper bounds, one must know the first many $p$ for which $\lambda\_f(p)=0$. Since we expect that no such $p$ exist, this amounts to verifying that $\lambda\_f(p)\neq0$ up to some large threshold using SAGE, MAGMA, etc. The arXiv post mentioned above appears to have some discussion on this.


For lower bounds, one needs an upper bound on $\pi\_f(x):=\#\{p\leq x\colon \lambda\_f(p)=0\}$ for all $x$ (use partial summation on the log of the product). Thus one needs an explicit version of Serre's density zero result, like the one given in Theorem 1.3 of the arXiv post mentioned above.


