Data-Driven Detection of Complex Multiplication in Weight 2 Cusp Forms

We introduce a machine learning approach for detecting Complex Multiplication (CM) in weight 2 newforms using a dataset of 53,779 modular forms from the LMFDB. By combining prime-indexed Fourier coefficients a_p for 25 primes up to 97 with 11 Sato-Tate moments M_2(d) and standardized ratios, we achieve F1=0.900 and precision=0.973 on an 80/20 held-out test set using Gradient Boosting Machines (GBM). Our contribution reveals M₄/M₂ as the most discriminative feature (importance 0.157), with trace coefficients at p=23, 41, and 7 contributing significantly. We find CM forms represent only 0.40% of the dataset (213/53,779), presenting a challenging class imbalance problem. Our results demonstrate that CM is learnable from small-dimensional feature sets without feature selection, providing a scalable alternative to Elliptic Curve analysis.