Conjectural quadratic-factor counts for the Hasse invariant of $E_7$

Let $l>7$ be prime with $l\equiv1,6\pmod7$. Let $\hat H_{7,l}(x)$ be the Hasse-invariant polynomial, and let $N_2$ count irreducible quadratic factors $x^2+ax+b$ dividing it modulo $l$ and satisfying $B(a,b)\equiv0\pmod l$, where

$$B(x,y)=x^3+(-5y+8)x^2+(-8y^2+6y+5)x-y^3-5y^2+8y-1.$$

Equivalently, $a\equiv(\alpha-1)b-\alpha\pmod l$ for some root $\alpha\in\mathbb F_l$ of $x^3-8x^2+5x+1\equiv0\pmod l$. Conjectural quadratic-factor formula. If $l\equiv1\pmod7$, then

$$N_2=\begin{cases}\frac32h(-7l)-\left(1-\left(\frac{-3}{l}\right)\right),&l\equiv1\pmod8,\\3h(-7l)-\left(1-\left(\frac{-3}{l}\right)\right),&l\equiv5\pmod8,\\frac34h(-7l)-\left(1-\left(\frac{-3}{l}\right)\right),&l\equiv3\pmod4.\end{cases}$$

If $l\equiv6\pmod7$, then

$$N_2=\begin{cases}\frac32h(-7l)-\left(4-\left(\frac{-3}{l}\right)\right),&l\equiv1\pmod8,\\3h(-7l)-\left(4-\left(\frac{-3}{l}\right)\right),&l\equiv5\pmod8,\\frac34h(-7l)-\left(4-\left(\frac{-3}{l}\right)\right),&l\equiv3\pmod4.\end{cases}$$

The claim gives a class-number formula for this constrained family of irreducible quadratic factors.