Indices of nilpotency in certain spaces of modular forms

We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level $N$ with integer coefficients modulo primes $p$ for $(p,N)\in \{(3,1),(5,1),(7,1),(3,4)\}$. In these settings, we prove upper bounds on certain indices of nilpotency. As an application of our bounds, we prove infinite families of congruences for $p^t$-core partition functions modulo $p$ for $p\in \{3,5,7\}$ and $t\ge 1$, and we prove an infinite family of congruences modulo $3$ for the $r$th power partition function, $p_r(n)$, when $r=12k$ with $\gcd (k,6)=1$. We also include conjectures on a function which quantifies degree lowering on powers of the Delta function by the relevant Hecke operators in these settings, and on the index of nilpotency relative to a modification of this degree-lowering function.