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 $s=2$. 



Lower bounds for Petersson inner products of cuspforms with integral Fourier coefficients
It's not hard to see that the answer to (a) is yes. There is a basis of $S\_2^{\textrm{new}}(\Gamma\_0(N))$ consisting of newforms. These newforms come into Galois orbits $\{f^\sigma\}\_{\sigma}$. Here $\sigma$ runs through the embeddings of $K\_f$ into $\mathbf{C}$, where $K\_f$ is the field generated by the Fourier coefficients of $f$. A basic fact is that the $\mathbf{C}$-span of a given Galois orbit is generated by cusp forms with integral coefficients. This follows from considering the forms $\sum\_{\sigma} \sigma(a) f^\sigma$ where $a$ runs through a $\mathbf{Z}$-basis of the ring of integers of $K\_f$.


It follows that if the newform $f$ has integral coefficients, then its orthogonal complement $f^\perp$ is generated by cusp forms with integral coefficients. Now consider the linear map $\lambda\_f : S\_2(\Gamma\_0(N),\mathbf{Z}) \to \mathbf{R}$ given by $\lambda\_f(g)=(f,g)$. By the previous remark, the kernel of $\lambda\_f$ has rank one less than the rank of $S\_2(\Gamma\_0(N),\mathbf{Z})$, which implies that the image of $\lambda\_f$ is of the form $c\_f \mathbf{Z}$ for some $c\_f >0$. Thus we can take for $c$ the minimum of all the $c\_f$. In fact $c\_f = (f,f)/m\_f$ for some integer $m\_f$ measuring the congruences of $f$ with other cusp forms.


