I'm trying to do an exercise in Diamond and Shurman's modular forms book which asks you to show that if $q$ is a prime, then the functions $f_q$ and $(1 + q)f_q - f_{q^2}$ form a Hecke eigenfunction basis of the Eisenstein subspace of $\Gamma_0(q^2)$. Here $$f_t = E_2(\tau) - tE_2(t\tau)$$ for $t \in \mathbb{N}$, where $E_2$ is the weight $2$ Eisenstein series $$E_2(\tau) = -\frac{1}{24} + \sum_{n \geq 1} \left(\sum_{d \mid n} d\right) e^{2\pi in \tau}.$$
I don't understand how this is possible. The dimension formulas for weight $2$ tell us that space $M_k(\Gamma_0(q^2)$ has dimension $g - 1 + \varepsilon_{\infty}$, where $g$ is the genus of $\Gamma_0(q^2)$ and $\varepsilon_{\infty}$ is the number of cusps of $\Gamma_0(q^2)$. On the other hand, the dimension of the cusp forms $S_2(\Gamma_0(q^2)$ is $g$. Then the Eisenstein subspace has diemsion $\varepsilon_{\infty} - 1$. The number of cusps for $\Gamma_0(q^2)$ is given by $$\sum_{d \mid q^2}\phi(\gcd(d,q^2/d)) = \phi(\gcd(1,q^2)) + \phi(\gcd(q,q)) + \phi(\gcd(q^2,1)) = 1 + q - 1 + 1 = q + 1,$$ so the Eisenstein subspace of $\Gamma_0(q^2)$ has dimension $q$. Then it can't possibly be spanned by two functions unless $q = 2$.