The precise 2-adic valuation conjecture for Eisenstein-series coefficients

Let $w_{a,k}$ be the coefficients defined by the expansion referred to as equation (w), let $\nu_2$ denote the 2-adic valuation, and let $s(k)$ denote the notation used in the source. For an even integer $k\geq 4$,

2-adic valuation conjecture.

$$\min_a\bigl(\nu_2(w_{a,k})\bigr)=\begin{cases}s(k)-2&\text{if }k\neq 2^j,\\0&\text{if }k=2^j. \end{cases}$$

This gives the precise expected value of the minimum 2-adic valuation of the coefficients in the polynomial expression for the Eisenstein series. The preceding theorem establishes only the lower bound that this minimum is nonnegative; the displayed equality is presented as a conjecture and is not resolved in the supplied text.