I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it!

Let $N\in\mathbb{N}$, let $q$ be a point in the open unit disc in the complex plane, and let $0\leq m \leq N$ be an integer. Some personal research I have been doing on iterated integrals of Eisenstein series has led me to consider certain series of the following form: $$S_m(N;q):=\sum_{k\geq 1}\frac{\sigma_N(k)}{k^N} k^m q^k,$$ where $\sigma_N(k):=\sum_{d\vert k} d^N$ is the $N$th power sum-of-divisors function.

There are two "extreme" cases that connect to well-known series: $m=0$ and $m=N$. In the $m=0$ case we recover a Lambert series $$S_0(N;q) = \sum_{k\geq 1}\frac{q^k}{k^N(1-q^k)}$$ and in the $m=N$ case (for $N\geq 3$ odd) we recover the nonconstant part of the $\mathbb{Q}$-normalised Eisenstein series $\mathbb{G}_{N+1}(\tau)$, with $q=\exp(2\pi i \tau)$ and $\tau$ in the upper half plane. Another connection to well-known objects comes from taking the limit $$\lim_{q\to 1}S_m(N;q) = \zeta(-m)\zeta(N-m).$$

In the cases $1\leq m\leq N-1$ I do not have a good name for these series, and I am not aware of them in the literature on modular forms/$L$-functions. I hope it is OK to ask: have these series been studied before, and are there any references for them if so? Is anything known about relations between the $S_m(N;q)$ (for a fixed $N$), or the dimension of the $\mathbb{Q}$-vector space generated by $\pi^m S_m(N;q)$ for $0\leq m\leq N$?

In fact, I am interested primarily not in the "functional" case, but with the specific case $q = e^{-2\pi}$. As an example of what I mean about relations in this case, Ramanujan's formula for $\zeta(N)$ in terms of a Lambert series gives a $\mathbb{Q}$-relation between $\zeta(N)$, $\pi^N$ and $S_0(N;e^{-2\pi})$ (at least for $N\equiv 3\pmod{4}$). I have been studying this case in my research and have found more relations between these numbers as well as $\pi^m S_m(N;e^{-2\pi})$ for larger $m$. If anything can be said about this case, I would be very grateful!