Congruences like Atkin's for the partition function

Let p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p( Q^3 \ell n+\beta )\equiv 0\pmod \ell where \ell and Q are prime and 5\leq \ell \leq 31; these lie in two natural families distinguished by the square class of 1-24\beta \pmod \ell. In recent decades much work has been done to understand congruences of the form p(Q^m\ell n+\beta )\equiv 0\pmod \ell. It is now known that there are many such congruences when m\geq 4, that such congruences are scarce (if they exist at all) when m=1, 2, and that for m=0 such congruences exist only when \ell =5, 7, 11. For congruences like Atkin’s (when m=3), more examples have been found for 5\leq \ell \leq 31 but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime \ell \geq 5, there are infinitely many congruences like Atkin’s in the first natural family which he discovered and that for at least 17/24 of the primes \ell there are infinitely many congruences in the second family.