nt.number theory – Does the Fourier expansion of the j-function have any prime coefficients?

Title asks it: Does the Fourier expansion of the j-function have any prime coefficients?

Congruences involving small primes rule out many candidates.

But
$$c_{71}=278775024890624328476718493296348769305198947=(353) (5533876049689057963) (142708463580969897033673)$$
so that might count as a near miss.

Up to $c_{6400}$ I find no primes. That said, though composite, none of $c_{1319},c_{1559},c_{1871},c_{2111},c_{2231},c_{3239},c_{3551},
c_{4271}, c_{4799}, c_{5471}$
has a factor less than 100.