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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 $(f,g)=\int_{X_0(N)}f(z)\overline{g(z)}dxdy$, $z=x+iy$, 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: If $f,g\in S_2(\Gamma_0(N))$ satisfy $a_n(f),a_n(g)\in\mathbb Z, n\geq 1$ and $(f,g)>0$, 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 $s=2$.

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 $s=2$.

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 $(f,g)=\int_{X_0(N)}f(z)\overline{g(z)}dxdy$, $z=x+iy$, 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: If $f,g\in S_2(\Gamma_0(N))$ satisfy $a_n(f),a_n(g)\in\mathbb Z, n\geq 1$ and $(f,g)>0$, 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 $f$ is 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 $s=2$.

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 $(f,g)=\int_{X_0(N)}f(z)\overline{g(z)}dxdy$, $z=x+iy$, 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: If $f,g\in S_2(\Gamma_0(N))$ satisfy $a_n(f),a_n(g)\in\mathbb Z, n\geq 1$ and $(f,g)>0$, 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 $s=2$.

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 $(f,g)=\int_{X_0(N)}f(z)\overline{g(z)}dxdy$, $z=x+iy$, 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: If $f,g\in S_2(\Gamma_0(N))$ satisfy $a_n(f),a_n(g)\in\mathbb Z, n\geq 1$ and $(f,g)>0$, 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.

The reason why I hope that $(*)$ could be true is the following. Suppose $f,g$ are as in the question. If $f,g$ satisfy in addition a ''Plancherel or Parseval type" formula $$(f,g)= c'\sum_{n=-\infty}^{\infty}a_n(f)\overline{a_n(g)}$$ for some constant $c'>0$, then $(*)$ holds with $c= c'$ since all $a_n(f),a_n(g)\in\mathbb Z$ and $(f,g)>0$. However, I have no idea if a ''Plancherel or Parseval type" formula as displayed above makes sense for modular forms.

My main interest is when $f$ is a newform for $\Gamma_0(N)$ with $a_1(f)=1$, but I don't know if this assumption simplifies things.

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 $(f,g)=\int_{X_0(N)}f(z)\overline{g(z)}dxdy$, $z=x+iy$, 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: If $f,g\in S_2(\Gamma_0(N))$ satisfy $a_n(f),a_n(g)\in\mathbb Z, n\geq 1$ and $(f,g)>0$, 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 $f$ is 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 $s=2$.