Theorem 1. Let \(H^{\mathrm{new}}(N,k)\) be a Hecke basis of the weight \(k\) cusp newforms for \(\Gamma_0(N)\) with trivial character, normalized to have leading coefficient \(1\). For each such \(f\), let \(\varepsilon(f)\in\{\pm1\}\) be its root number, \(a_f(p)\) its \(p\)-th Fourier coefficient, and \(\lambda_f(p)=a_f(p)/p^{(k-1)/2}\). Let \(X,Y\to\infty\) and let \(P\to\infty\) through the primes, with \(Y=(1+o(1))X^{1-\delta_2}\) and \(P\ll X^{1+\delta_1}\), where \(0<\delta_1<1/11\) and \(2\delta_1<\delta_2<\tfrac{1}{13}(4-18\delta_1)\). Put \(y=P/X\). Then $$\frac{\displaystyle\sum^{\square}_{N\in[X,X+Y]}\ \sum_{f\in H^{\mathrm{new}}(N,k)}\sqrt{P}\,\lambda_f(P)\,\varepsilon(f)}{\displaystyle\sum^{\square}_{N\in[X,X+Y]}\ \sum_{f\in H^{\mathrm{new}}(N,k)}1} =\mathcal{M}_k(y)+O_\epsilon\!\left(X^{-\delta'+\epsilon}+\frac{1}{P}\right),$$ where \(\sum^{\square}\) runs over squarefree \(N\), \(\delta'>0\) depends only on \(\delta_1,\delta_2\), and \(\mathcal{M}_k\) is the weight \(k\) murmuration density $$\mathcal{M}_k(y)=\frac{\alpha(-1)^{k/2-1}}{k-1}\sum_{1\le r\le 2\sqrt{y}}\nu(r)\sqrt{4y-r^2}\,U_{k-2}\!\left(\frac{r}{2\sqrt{y}}\right)+\frac{\beta}{k-1}\sqrt{y}-\gamma\,\delta_{k=2}\,y.$$ Here \(U_{k-2}(\cos\theta)=\sin((k-1)\theta)/\sin\theta\) is a Chebyshev polynomial, $$\nu(r)=\prod_{p\mid r}\left(1+\frac{p^2}{p^4-2p^2-p+1}\right),$$ $$\alpha=2\pi\prod_p\frac{p^4-2p^2-p+1}{p^4-2p^2+p},\qquad \beta=2\pi\prod_p\frac{p^3+p^2-1}{p(p^2+p-1)},\qquad \gamma=12\prod_p\frac{p(p+1)}{p^2+p-1}.$$
The film shows weight \(k=2\).