Denote $\operatorname{dim}(M_k(\Gamma_0(N)))$ by $m(k,N)$ and $\operatorname{dim}(S_k(\Gamma_0(N)))$ by $s(k,N)$. Let $N$ any positive multiple of $4$ and $j \ge 1$. $$ a(N) := \frac1j \left(m \left(\frac{4j+1}{2},N\right) +s\left(\frac{4j+3}{2}, N\right) \right) $$ For $N = 4,8,12, \ldots$ this gives the first table in H. Cohen and J. Oesterlé (see below) and was registered by Steven Finch in OEIS A159634.

Recently E. P. Herrero conjectured the identity $$ a(N) = \frac{1}{3}\Psi\left(\frac{N}{2}\right). \qquad ({\Psi \text{ denotes the Dedekind Psi function.)}} $$

The question is: Are -- apart from numerical evidence -- reasons known which support this conjecture?

*Addition:*

Meanwhile W. Meeussen made a related conjecture with regard to $$ b(N) = m\left(\frac{k}{2}, N\right) - s\left(\frac{k}{2}, N\right), $$

where $N$ is any positive multiple of 4 and $k \ge 5$ is odd.

Let [] denote the Iverson brackets, $\varphi$ Euler's totient function and $$ \chi(n) = \sum_{d|n} \varphi \left(\operatorname{gcd} \left(d, \frac {n}{d} \right) \right). $$

Then Meeussen conjectured $$ b(N) = 2 \chi\left(\frac{N}{2}\right) - \frac 12 \chi\left(\frac{N}{2}\right)\, \left[ \frac{N}{2}+2 \bmod 4 = 0\right] . $$

This is OEIS sequence A159633.

Ref.:

H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78.

S. R. Finch, Primitive Cusp Forms, April 27, 2009.