Let $N\geq 1$ be an integer and let $S_2(\Gamma_0(N))$ be the cusp forms of weight 2 for the usual congruence subgroup $\Gamma_0(N)\subset SL_2(\mathbb Z)$. Let $a_n(f)$ denote the n-th Fourier coefficient of $f\in S_2(\Gamma_0(N))$, $n\geq 1$. Let $X_0(N)$ be a smooth projective model over $\mathbb Q$ of the modular curve associated to $\Gamma_0(N)$ and let $$(f,g)=\int_{X_0(N)}f(z)\overline{g(z)}dxdy,$$ $z=x+iy$, be the Petersson inner product of $f,g\in S_2(\Gamma_0(N))$.

Question (a) Does there exist a constant $c>0$, depending at most on $\Gamma_0(N)$ (or $X_0(N)$), with the following property: Suppose $f\in S_2(\Gamma_0(N))$ is an eigenform with $a_1(f)=1$ which lies in the new part, $g\in S_2(\Gamma_0(N))$ and $a_n(f),a_n(g)\in\mathbb Z, n\geq 1$. If $(f,g)>0$, then $$(f,g)\geq c.$$

(b) Can one find such a constant $c>0$ with an explicit dependence on $N$.

(c) Can one find such a constant $c>0$ which is absolute.

My main interest is when $f$ in the above question is in addition a newform for $\Gamma_0(N)$ with $a_1(f)=1$, but I don't know if this assumption simplifies things. Further, David Loeffler mentions below that $(f,g)$ is related to a residue of a certain $L$-series at $s=2$.