I am trying to prove the sums of 3 squares formula in Cohen's 1975 paper 'Sums Involving the Values at Negative Integers of L-Functions of Quadratic Characters'. That is, for $H(N)$ the Hurwitz class number, and $r_3(n)$ the number of ways of representing $n$ as a sum of 3 squares. we have: \begin{equation} r_3(4N)=r_3(N), \end{equation} \begin{equation} r_3(N)=0 \text{ if N $\equiv7$ mod $8$ }, \end{equation} \begin{equation} r_3(N)=12H(4N) \text{ if N $\equiv1,2,5$ or 6 mod $8$}, \end{equation} \begin{equation} r_3(N)=24H(N) \text{ if N $\equiv 3$ mod $8$}. \end{equation} The first $2$ of these have elementary proofs. It is the other $2$ I am stuck on.
Attempt: Use $a=8$ in corollary $3.4$ as suggested. According to the corollary, the function \begin{equation} \mathscr{H}_1(z,a,b)=\sum_{N\equiv b \text{ mod } a}H(N)q^N \end{equation} is a modular form of weight $3/2$ and character $1$ for $\Gamma_0(64)$ with $a=8$ and $b=1,2,3,5,6$. Analagously the sum of 4 squares problem, I would expect to use the dimension, say $d$, of $M_{3/2}(\Gamma_0(64),1)$, taking $d+1$ linearly independent elements of $M_{3/2}(\Gamma_0(64),1)$ including \begin{equation} \theta^3(z)=1+\sum_{N\geq1}r_3(N)q^N. \end{equation} (The function $\theta(z)$ is well known to be a modular form of weight 1/2, character $1$, for $\Gamma_0(4)$ and therefore for $\Gamma_0(64)$, and its cube will have weight $3/2$). Then solving a linear system of equations to obtain the answer.
We have $\mathscr{H}_1(z,8,b)=0$ for $b=1,2,5$ and $6$, so it seems to make sense that $r_3(N)$ has a different formula for $N\equiv3$ mod $8$ than for $N\equiv1,2,5,6$ mod $8$. But I don't know how to get any further.