We determine the average size of the eigenvalues of the Hecke operators. To measure average size, we use the quadratic mean. We consider both the vertical and the horizontal perspective.

§ INTRODUCTION

Let $T'_m(N,k) := T_m(N,k) / m^{(k-1)/2}$ denote the $m$-th normalized Hecke operator over $S_k(\Gamma_0(N))$. When $N$ and $k$ are clear from the context, we will just write $T'_m$.
In this paper, we study the average size of the eigenvalues for these Hecke operators.

One could interpret the notion of average size here in two different ways: vertically and horizontally.
In Section <ref>, we consider the vertical perspective; where $m$ is fixed and $N$ and $k$ vary. For $m$ fixed, consider $N$ coprime to $m$ and $k \geq 2$ even. Also, let $s(N,k) = \dim S_k(\Gamma_0(N))$ and  $\lambda_1,\cdots, \lambda_{s(N,k)}$ be the eigenvalues of $T'_m(N,k)$.  Then to measure the average size of the eigenvalues of $T'_m(N,k)$, we use the quadratic mean,

\begin{align} \label{eqn:Av-m-def}
\Av_m(N,k):=\sqrt{\frac{1}{s(N,k)}\sum_{i=1}^{s(N,k)} \lambda_i^2}.
\end{align}

We note that since $N$ is coprime to $m$, the $\lambda_i$ are real <cit.>, and hence the above formula actually represents the average size.
We also note that the quadratic mean is more natural to study than, for example, the arithmetic mean of the absolute values of the $\lambda_i$; squares are much nicer to work with than absolute values.

In Theorem <ref>, we will determine the behavior of $\Av_m(N,k)$ as $N+k \to \infty$. In the following, $\sigma_1(m)$ denotes the sum of divisors function, $\sigma_1(m) = \sum_{d\mid m} d$.

Let $m$ be fixed. Then for $N$ coprime to $m$ and $k \geq 2$ even,

$$\Av_m(N,k) \longrightarrow \sqrt{\frac{\sigma_1(m)}{m}} \quad \textnormal{as} \quad N+k \longrightarrow \infty.$$

Our method is effective; we have explicit bounds determining exactly how fast $\Av_m(N,k)$ converges to $\sqrt{\frac{\sigma_1(m)}{m}}$.

As an example, in Section <ref>, we use these explicit bounds to

classify precisely when $\Av_2(N,k) \le 1$.

In Section <ref>, we consider the horizontal perspective; where $N$ and $k$ are fixed and $m$ varies. From this perspective, it is more natural to think of the eigenvalues of $T'_m(N,k)$ as the normalized Fourier coefficients of newforms. Let $f$ be a newform of weight $k$ and level $N$. Let $a'_f(m) := a_f(m) / m^{(k-1)/2}$ denote the normalized Fourier coefficients of $f$, i.e. the eigenvalues of $T'_m(N,k)$ associated to $f$. Then to measure the average size of the $a'_f(m)$, we again use the quadratic mean,

\begin{align} \label{eqn:Avf-def}
\Av_f(x) := \sqrt{ \frac{1}{x} \sum_{m \leq x} a'_f(m)^2 }.
\end{align}

We note that since $f$ is a newform, the $a'_f(m)$ are real <cit.>, and hence the above formula actually represents the average size.
In Theorem <ref>, we determine the behavior of $\Av_f(x)$ as $x \to \infty$.

Let $f$ be a newform of level $N$ and weight $k$. Then

$$\Av_f(x) \longrightarrow \sqrt{\frac{12 \cdot (4\pi)^{k-1}}{(k-1)!}} \cdot \Vert f \Vert \quad \textnormal{as} \quad x \longrightarrow \infty.$$

Here, $\Vert f \Vert$ denotes the Petersson norm, $\Vert f \Vert = \sqrt{\langle f,f \rangle}$.

The idea of this theorem is not new. Although perhaps not stated in these terms, others have considered equivalent questions (e.g. <cit.>, <cit.>, <cit.>). However, we state this theorem here for completeness.

Finally, in Section <ref>, we discuss our results and propose a lower bound on the size of the Fourier coefficients of newforms, which mimics the Atkin-Serre conjecture.

§ THE VERTICAL PERSPECTIVE

In this section, we take the vertical perspective. That is, for $m$ fixed, we study $\Av_m(N,k)$ as $N+k \to \infty$. Note that $s(N, k) = 0$ for only finitely many pairs $(N,k)$ <cit.>, so $\Av_m(N,k)$ here is well-defined for $N + k$ sufficiently large.

Now, the case of $m=p$ prime follows from already known results. In <cit.>, Serre showed that assuming $p \nmid N$, the asymptotic distribution of the eigenvalues of $T_p(N,k)$ is given by $\displaystyle \mu_p = \frac{p+1}{\pi} \frac{(1-t^2/4)^{1/2} }{ (p^{1/2} + p^{-1/2})^2 - t^2 }  \,dt$.
This means that

\begin{align}
\lim_{N+k \to \infty} \Av_p(N,k)^2
&=   \lim_{N+k \to \infty} \frac{1}{s(N, k)} \displaystyle \sum_{i=1}^{s(N,k)} \lambda_i^2  \\
&=   \int_{-2}^2 \frac{p+1}{\pi} \frac{(1-t^2/4)^{1/2} }{ (p^{1/2} + p^{-1/2})^2 - t^2 }  \cdot t^2 \,dt \\
&=  \frac{p+1}{p},
\end{align}

and hence

\begin{equation}
\Av_p(N,k) \longrightarrow \sqrt{\frac{p+1}{p}} \qquad \textnormal{as} \qquad N+k \longrightarrow \infty. \label{eqn:Av-asymp-prime}
\end{equation}

However, this approach will not work for general $m$ because Serre's methods do not produce a measure $\mu_m$ for composite $m$. Instead, we compute the asymptotics of $\Av_m(N,k)$ directly by writing $\Av_m(N,k)$ in terms of traces of certain Hecke operators.

Let $\lambda_1, \ldots, \lambda_{s(N,k)}$ be the eigenvalues of $T'_m$. Then using the Hecke operator composition formula <cit.>,

\begin{align}
\sum_{i=1}^{s(N,k)} \lambda_{i}^2
&= \Tr \lrp{{T'_m}^2} = \Tr \sum_{d\mid m}  T'_{m^2/d^2} = \sum_{d\mid m}  \Tr T'_{m^2/d^2}.
\end{align}

Thus by (<ref>),

\begin{equation} \label{eqn:Av-m-trace-formula}
\Av_m(N,k) = \sqrt{\frac{1}{s(N, k)}  \sum_{d\mid m} \Tr T'_{m^2/d^2}}.
\end{equation}

We now cite a lemma to estimate the traces of Hecke operators appearing in this formula.
In the following, all big-$O$ notation is vertical, i.e. with respect to $N$ and $k$.

Let $m$ be a fixed perfect square. Then for $N$ coprime to $m$ and $k \geq 2$ even,

\begin{align} \label{eqn:trace-estimate-perfect-square}
\Tr T'_m = \frac{k-1}{12} \psi(N) \frac{1}{\sqrt{m}} + O(N^{1/2+\varepsilon}) \qquad \text{for any } \varepsilon > 0.
\end{align}

Here, $\psi(N)$ is the multiplicative function $\psi(N) = N \prod_{p\mid N} (1 + 1/p)$. Note that $\psi(N) \geq N$. We then use this lemma to determine the asymptotic behavior of $\Av_m(N,k)$.

Let $m$ be fixed. Then for $N$ coprime to $m$ and $k \geq 2$ even,

$$\Av_m(N,k) \longrightarrow \sqrt{\frac{\sigma_1(m)}{m}} \quad \textnormal{as} \quad N+k \longrightarrow \infty.$$

By (<ref>) and Lemma <ref>,

\begin{align}
\Av_m(N,k)^2 &= \frac{1}{s(N, k)} \sum_{d\mid m} \Tr T'_{m^2/d^2}  \\
&= \frac{1}{s(N, k)} \sum_{d\mid m}  \lrp{ \frac{k-1}{12} \psi(N)  \frac{d}{m} + O(N^{1/2+\varepsilon}) } \\
&= \frac{\sigma_1(m)}{m} \cdot  \frac{\frac{k-1}{12} \psi(N)}{s(N, k)}  + \frac{O(N^{1/2+\varepsilon})}{s(N, k)}. \label{eqn:Av-m-asymp-temp}
\end{align}

Note from Lemma <ref> that $s(N, k) = \Tr T'_1 = \frac{k-1}{12} \psi(N) + O(N^{1/2+\varepsilon})$.
Thus as $N+k \longrightarrow \infty$,

\begin{align}
\frac{\frac{k-1}{12} \psi(N)}{s(N, k)}  \longrightarrow 1 \qquad\qquad \text{and} \qquad\qquad \frac{O(N^{1/2+\varepsilon})}{s(N, k)} \longrightarrow 0.
\end{align}

So by (<ref>), $\Av_m(N,k)^2 \longrightarrow \frac{\sigma_1(m)}{m}$, proving the desired result.

§ A CLASSIFICATION OF WHEN AV_2(N,K)

Our method is effective; we have explicit bounds for all of the error terms in the proof of Theorem <ref>, which allows us to determine exactly how fast $\Av_m(N,k)$ converges to $\sqrt{\frac{\sigma_1(m)}{m}}$.
The fact that $\Av_m(N,k) \longrightarrow \sqrt{\frac{\sigma_1(m)}{m}}$ means that for any $\alpha < \sqrt{\frac{\sigma_1(m)}{m}}$, we will have $\Av_m(N,k) \leq \alpha$ for only finitely many pairs $(N,k)$. We take $m=2,\, \alpha=1$ in particular, and compute the complete list of pairs $(N,k)$ for which $\Av_2(N,k) \leq 1$.

Consider $N\geq1$ odd and $k\geq2$ even such that $\dim S_k(\Gamma_0(N)) > 0$. Then $\Av_2(N,k) \leq 1$ only for the pairs $(N,k)$ given in Table <ref>.

From (<ref>),

\begin{align}
\Av_2(N,k) &= \sqrt{1 + \frac{\Tr T_{4}'}{ s(N,k)}}.
\end{align}

So $\Av_2(N,k) \le 1$ precisely when $\Tr T_4' \le 0$. By Lemma <ref> we have that $\Tr T'_4 = \frac{k-1}{24} \psi(N) + O(N^{1/2+\varepsilon})$ for any $\varepsilon>0$. And in fact, <cit.> gave the explicit lower bound

\begin{align} \label{eqn:T4-lower-bound-temp}
\Tr T_4'(N,k) &\geq  \psi(N) \lrb{\frac{k-1}{24} - \lrp{14 \cdot \frac{2^{\omega(N)}}{\psi(N)} + \frac12 \frac{2^{\omega(N)} \sqrt{N}}{\psi(N)}} }.
\end{align}

Then for $N \geq 150,\!000$, <cit.> gives the explicit bounds

\begin{align} \label{eqn:theta-i-explicit-bounds}
\frac{2^{\omega(N)}}{\psi(N)} \leq 0.000147 \qquad \text{and} \qquad \frac{2^{\omega(N)} \sqrt{N}}{\psi(N)} \leq 0.0607,
\end{align}

which means that

\begin{align}
\lrp{14 \cdot \frac{2^{\omega(N)}}{\psi(N)} + \frac12 \frac{2^{\omega(N)} \sqrt{N}}{\psi(N)}} \leq 14 \cdot 0.000147 + \frac12 \cdot 0.0607 < \frac{1}{24} \leq \frac{k-1}{24}.
\end{align}

This means that by (<ref>), $\Tr T'_4(N,k) > 0$ for all $N \geq 150,\!000$. Then for each of the finitely many remaining values of $N$, we will also have $\frac{k-1}{24} > \lrp{14 \frac{2^{\omega(N)}}{\psi(N)} + \frac12 \frac{2^{\omega(N)} \sqrt{N}}{\psi(N)}}$, and hence $\Tr T'_4(N,k) > 0$, for sufficiently large $k$.
Finally, we check the finitely many remaining pairs $(N,k)$ by computer, which yields the complete list given in Table <ref>. See <cit.> for the code.

\begin{equation}
\setlength{\arraycolsep}{2mm}
\begin{array}{|c|c||c|c||c|c||c|c||c|c|}
\hline
\multicolumn{10}{|c|}{\makecell{
\text{The complete list of pairs $(N,k)$ for which $\Av_2(N,k) \leq 1$.  }
}} \\
\hline
(N,k) & \Av_2  &  (N,k) & \Av_2  &  (N,k) & \Av_2  &  (N,k) & \Av_2  &  (N,k) & \Av_2  \\
\hline
\hline
(1, 12) & \frac{3}{8}\sqrt{2} & (3, 8) & \frac{3}{8}\sqrt{2} & (17, 2) & \frac{1}{2}\sqrt{2} & (37, 2) & 1 & (63, 2) & \frac{3}{10}\sqrt{10} \\
\hline
(1, 20) & \frac{57}{128}\sqrt{2} & (5, 6) & \frac{1}{4}\sqrt{2} & (19, 2) & 0 & (45, 2) & \frac{1}{2}\sqrt{2} & (81, 2) & \frac{1}{2}\sqrt{3} \\
\hline
(1, 22) & \frac{9}{64}\sqrt{2} & (7, 4) & \frac{1}{4}\sqrt{2} & (21, 2) & \frac{1}{2}\sqrt{2} & (49, 2) & \frac{1}{2}\sqrt{2}  & \multicolumn{2}{c}{~} \\
\cline{1-8}
(1, 26) & \frac{3}{512}\sqrt{2} & (9, 4) & 0 & (23, 2) & \frac{1}{2}\sqrt{3} & (53, 2) & 1  & \multicolumn{2}{c}{~} \\
\cline{1-8}
(1, 30) & \frac{3}{512}\sqrt{26687} & (11, 4) & \frac{1}{2}\sqrt{2} & (27, 2) & 0 & (57, 2) & \frac{3}{10}\sqrt{10}  & \multicolumn{2}{c}{~} \\
\cline{1-8}
(1, 46) & \frac{3}{2^{16}}\sqrt{468559893} & (15, 2) & \frac{1}{2}\sqrt{2} & (31, 2) & \frac{1}{2}\sqrt{3} & (61, 2) & 1
& \multicolumn{2}{c}{~} \\
\cline{1-8}
\end{array}
\end{equation}

§ THE HORIZONTAL PERSPECTIVE

In this section, we take the horizontal perspective. That is, for $f$ fixed, we study $\Av_f(x)$ as $x \to \infty$, as defined in (<ref>). Recall that $f$ here is a newform of weight $k$ and level $N$, and that $a'_f(m)$ denotes the $m$-th normalized Fourier coefficient of $f$.

We note that, similarly to the vertical perspective, the average size of the prime-indexed normalized Fourier coefficients follows from already known results.
When $f$ is CM, the distribution of the $a_f'(p)$ tends to the measure $\mu_{CM} = \lrp{\frac{1}{2}\delta_0(t) + \frac{1}{4\pi}\frac{1}{\sqrt{1 - t^2/4}}} dt$, where $\delta_0$ is the Dirac delta function <cit.>, <cit.>. Consequently,

\begin{align} \label{eqn:avg-afp2-cm}
\lim_{x \to \infty} \frac{1}{\#\{ \text{primes } p \leq x\} } \sum_{  p \leq x }  a'_f(p)^2
&= \int_{-2}^2 \lrp{\frac{1}{2}\delta_0(t) + \frac{1}{4\pi}\frac{1}{\sqrt{1 - t^2/4}} } \cdot t^2 \,dt = 1.
\end{align}

When $f$ is non-CM, Barnet-Lamb, Geraghty, Harris, and Taylor showed that the distribution of the $a'_f(p)$ tends to the Sato-Tate measure $\displaystyle \mu_{ST} = \frac{1}{\pi} \sqrt{1-t^2/4}  \,dt$ as $p \to \infty$ <cit.>, <cit.>. Consequently,

\begin{align} \label{eqn:avg-afp2}
\lim_{x \to \infty} \frac{1}{\#\{ \text{primes } p \leq x\} } \sum_{  p \leq x }  a'_f(p)^2
&= \int_{-2}^2 \frac{1}{\pi} \sqrt{1-t^2/4} \cdot t^2 \,dt = 1.
\end{align}

In this sense, the average size of the $a'_f(p)$ is $\sqrt{1} = 1$, regardless of whether $f$ is CM or not. Interestingly, if we consider the general $r$-th mean (as opposed to just using the quadratic mean at $r=2$), then

$$\lim_{x \to \infty} \lrp{ \frac{1}{\#\{ \text{primes } p \leq x\} } \sum_{  p \leq x }  \lrabs{a'_f(p)}^r}^{1/r}$$

agrees for CM and non-CM newforms only at $r = 2$.

The average size of all the normalized Fourier coefficients of $f$ also follows from already known results. Recall that we are interested in determining

\begin{align}
\lim_{x\to\infty} \Av_f(x) =  \sqrt{ \lim_{x \to \infty} \frac{1}{x}  \sum_{  m \leq x }  a'_f(m)^2}  .
\end{align}

This problem reduces to determining the asymptotic behavior of $\sum_{  m \leq x }  a'_f(m)^2$. By the Weiner-Ikehara theorem <cit.>, this in turn reduces to determining the residue of the Dirichlet series $L_f(s) := \sum_{m = 1}^\infty a'_f(m)^2 m^{-s}$ at $s=1$. This Dirichlet series is closely related to the symmetric square $L$-function, and has already been studied extensively; in particular, its residues are known.
From <cit.>, we have that $L_f(s)$ can be extended meromorphically to the entire complex plane, and that it has a simple pole at $s=1$ of residue $\Res_{s=1} L_f(s) = \frac{12 \cdot (4\pi)^{k-1}}{(k-1)!} \langle f,f \rangle$. This fact was originally proven by Rankin <cit.> in order to show that

\begin{align} \label{eqn:avg-tau-unnorm}
\sum_{m\leq x} \tau(m)^2 \ \sim \ \lrp{\frac{1}{12}  \Res_{s=1} L_\Delta(s)} \cdot x^{12} = \frac{(4\pi)^{11}}{11!} \langle \Delta,\Delta \rangle \cdot x^{12}
\end{align}

In our case, we are interested in something slightly different, the normalized Fourier coefficients. By the Weiner-Ikehara Theorem, we have $\sum_{m\leq x} a'_f(m)^2 \ \sim \ \lrp{\Res_{s=1} \, L_f(s)} \cdot x$. This means that
\begin{align}
\lim_{x\to\infty} \Av_f(x)^2 = \lim_{x\to\infty} \frac{1}{x} \sum_{m\leq x} a'_f(m)^2 = \Res_{s=1} \, L_f(s) = \frac{12 \cdot (4\pi)^{k-1}}{(k-1)! } \langle f,f \rangle,
\end{align}

verifying Theorem <ref>.

As an example, we use this theorem to give the average size of the normalized Ramanujan $\tau$ function. Note the distinction between the average size and the constant given in (<ref>) by Rankin, which is due to our normalization of the Fourier coefficients.

The average size of the normalized Ramanujan $\tau$ function is given by

\begin{align}
\lim_{x\to\infty} \Av_\Delta(x) = \sqrt{\frac{12 \cdot (4\pi)^{11}}{(11)!} } \Vert \Delta \Vert = 0.619745...
\end{align}

For this numerical value, we used $\Vert \Delta \Vert = 0.001017527... $ from <cit.>.

§ DISCUSSION

For any newform $f$, recall that the optimal

upper bound on the size of $a'_f(m)$ (for composite $m$) is a natural generalization of the optimal upper bound on the size of $a'_f(p)$ (for prime $p$). We have Deligne's bound $\lrabs{a'_f(p)} \leq 2$, and the natural generalization $\lrabs{a'_f(m)} \leq \sigma_0(m)$.
Additionally, we have from (<ref>), (<ref>), and Theorem <ref> that the average size of the $a'_f(m)$ is a (nonzero) constant factor of the average size of the $a'_f(p)$.

For these reasons, we also conjecture that the Fourier coefficients of $f$ should have a lower bound that resembles a lower bound on the prime-indexed Fourier coefficients. In particular, we propose Conjecture <ref>, which mimics the Atkin-Serre conjecture <cit.> on prime-indexed Fourier coefficients.

Let $f$ be a

newform of weight $k \geq 4$ and level $N$.

Then for all $\varepsilon > 0$, there exists a constant $c_\varepsilon > 0$ such that for all $m\ge1$, if
$a_f(m) \neq 0$, then $\lrabs{a_f(m)} \geq c_\varepsilon m^{(k-3)/2 - \varepsilon}$.

This conjecture aligns with our numerical computations <cit.>.
We note that contrary to the classical Atkin-Serre conjecture, we must account for the fact that it is possible to have $a_f(m) = 0$ for infinitely many $m$. By multiplicativity, even if $a_f(p) = 0$ for only one prime $p$, then $a_f(m) = 0$ for infinitely many $m$. Restricting to $m$ such that $a_f(m) \neq 0$ also makes it possible to remove the “non-CM" condition from classical Atkin-Serre.

§ ACKNOWLEDGEMENTS

This research was supported by NSA MSP grant H98230-24-1-0033.