I've posted a short note to the arxiv called "On Murmurations and Trace Formulas". (LD25c Knowl) This is an expanded version of my talk On Murmurations and Trace Formulas that I gave at the Simons Center for Geometry and Physics workshop on murmurations last November. The organizers of that workshop gathered several of the talks together for a special issue of the International Journal of Data Science in the Mathematical Sciences.
Several of my coauthors and collaborators were giving talks at that workshop, prompting me to give more of a speculative talk looking forwards instead of the standard talk describing some past work. I have given "typical talks" on my work in murmurations before. See my talk at LSU in November 2024 for my most recent.
Here, I want to further discuss an aspect that I think is commonly confusing in talks and the literature: the issue of normalization.
Algebraic and Analytic Normalization
The first murmurations observed were of elliptic curves, noting oscillatory
behavior in averages of
More precisely, for any conductor
This leads to the patterns in the frequently-cited murmuration plot.This is made to follow the plot in Murmurations of elliptic curves, https://arxiv.org/pdf/2204.10140, by He, Lee, Oliver, and Pozdnyakov.

Implicitliy, the coefficients of the elliptic curves are normalized in the
standard "algebraic" way. That is, the coefficients
This normalization is natural with elliptic curves, but it appears that
murmuration phenomena appear for many families of
When looking for murmurations in other families of
Why do values work so well?
Let's examine the elliptic curve case more closely.
The coefficient
But we're not summing absolute values. As the signs are fluctuating randomly from one curve to the next, the sum should experience some cancellation.1 1We'll return to how much cancellation in a moment. The effect is that that the extra coefficient size mostly counteracts the sign-change cancellation, giving a well-behaved overall sum.
Is normalization necessary?
Having said that, what if we just don't use the algebraic normalization? What if we use only the analytic normalization? On the same data as the original murmuration plot, we get the following plot.

This clearly shows murmuration phenomena and has the same broad structure — with the one notable exception that the overall murmuration is decreasing in magnitude.
In my weight
In short, I think there is an overemphasis on normalization.
Is the algebraic normalization the "best"?
Let's look closer at the sums.
Suppose there are
Then we look at
That is, we should expect the murmuration function to grow slightly in
If we want constant size murmurations, then this suggests the "correct"
normalization of looking at
This would give the following.

In principle, it could be informative to compare to the standard

The expected growth
The size of the family played an important role in this discussion!
The "correct" normalization for modular forms of weight
But the point remains that perfectly normalizing the coefficients so that the murmuration function is constant on average isn't important!
Leave a comment
Info on how to comment
To make a comment, please send an email using the button below. Your email address won't be shared (unless you include it in the body of your comment). If you don't want your real name to be used next to your comment, please specify the name you would like to use. If you want your name to link to a particular url, include that as well.
bold, italics, and plain text are allowed in
comments. A reasonable subset of markdown is supported, including lists,
links, and fenced code blocks. In addition, math can be formatted using
$(inline math)$
or $$(your display equation)$$
.
Please use plaintext email when commenting. See Plaintext Email and Comments on this site for more. Note also that comments are expected to be open, considerate, and respectful.
Comments (2)
2025-06-09 Em
Where is the data for the plots? Can I download it?
2025-06-12 David Lowry-Duda
The data is from the LMFDB. I made these plots using code similar to that in my note on distinguishing sets of coefficients. If you're interested in playing with the data, I suggest trying out the api options. Let me (or other LMFDB people) know if you have trouble and we can help out.