Let \(j(z) = q^{-1} + 744 + \sum_{n=1}^\infty C(n) q^n\) (with \(q= e^{2\pi\text{i}z}\) as usual) be Klein’s modular function. Its Fourier-coefficients \(C(n)\) satisfy some congruence relations. As J. Lehner has shown in [Discontinuous groups and automorphic functions, American Mathematical Society (1964; Zbl 0178.42902)] we find \(C(n)\equiv 0 \pmod {2^{3a+8}3^{2b+3}5^{c+1}7^d}\) if \(2^a3^b5^c7^d\) divides \(n\). In particular, if \(n\) is divisible by some power of 2, \(C(n)\) is divisible by a higher power of 2. Another type of condition which implies congruences modulo powers of 2 is being introduced in the paper under review. It is shown that for any \(t\in {\mathbb N}\) there is some \(c\in {\mathbb N}\) such that \(C(n)\) is divisible by \(2^t\) if there are \(c\) different primes \(p_1,\dots ,p_c\geq 3\) such that \(p_i\) divides \(n\) but does not divide \(n/p_i\). This shows that for almost all values of \(n\) (Dirichlet density) \(C(n)\) is divisible by \(2^t.\)
Similar results for partition numbers and representation numbers of some quadratic forms are given, as well as for Fourier coefficients of Kohnen newforms. The type of result resembles and generalizes results of the first author and B. Gordon [Ramanujan J. 1, No. 1, 25–34 (1997; Zbl 0907.11036)] on partition numbers. The proof uses congruences relating holomorphic and non-holomorphic modular forms and the topological nilpotency of Hecke operators acting on some spaces of 2-adic cusp forms as well as a list of 2-dimensional Galois-representations of the absolute Galois group of the field of rational numbers in characteristic 2 having small conductor.