Degree and nilpotence-index conjectures for the modified Hecke operator at 19 modulo 5

Let $p=5$, let $k\geq 1$ with $5\nmid k$, and let $D_{19}(k,5)$ denote the degree in $\Delta$ of $\Delta^k\mid T_{19}^{\prime}$ modulo $5$. Define $v_5(n)$ to be the largest power of $5$ dividing $n$, and let $S_{19}^{\prime}(k,5)$ be the least $t\geq 1$ for which the $t$th iterate of the degree-lowering function reaches $-\infty$. If $k=\sum a_i5^i$ and $s=a_0+5a_1$, define $c_t=S_{19}^{\prime}(5^t+1,5)-1$.

Degree and nilpotence-index conjecture. The degree $D_{19}(k,5)$ is given by the displayed residue-class formulas in the source; moreover, $c_0=0$, $c_1=2$, and

$$c_i=3c_{i-1}+2c_{i-2}\quad(i\geq2),$$

and

$$S_{19}^{\prime}(k,5)=S_{19}^{\prime}(s,5)+\sum_{i\geq2}a_ic_i.$$

Further,

$$S_{19}^{\prime}(k,5)=O(k^{\alpha}),\qquad \alpha=\log_5\left(\frac{3+\sqrt{17}}2\right)\approx0.7892,$$

and $N_{19}(\Delta^k,5)\leq S_{19}^{\prime}(k,5)$.