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Modular Form Murmurations

Patterns extend beyond elliptic curves to modular forms

f(z) = Σ aₙ qⁿ, q = e^(2πiz)

Modular form Fourier expansion

Even sign murmuration
Odd sign murmuration
Combined average

Beyond Elliptic Curves

After Zubrilina's proof, mathematicians including Bober, Booker, Min Lee, and Lowry-Duda showed murmurations exist in general modular forms and Dirichlet characters.

This universality suggests murmurations are fundamental to number-theoretic L-functions.