Eisenstein series of weight $2$ for in $M_2(\Gamma_0(N))$

Question

Let $q$ be prime. By dimension formulas, $S_2(\Gamma_0(q))$ is of dimension $g$, while $M_2(\Gamma_0(q))$ is of dimension $g+1$. So the space of Eisenstein series is one dimensional.

1. How to define this Eisenstein series?
2. For $p$ prime, what is the eigenvalue of the Hecke operator $T_p$ acting on this Eisenstein series? (I recall it should be $p+1$, but not sure).
3. Is it nonzero at both cusps, or just in one of them?