The paper establishes the new state-of-art in the accuracy analysis of Hutchinson's estimator for the matrix trace. Leveraging tools that have not been previously used in this context, particularly hypercontractive inequalities and results on concentration of sub-gamma distributions, we offer an elegant and modular analysis, as well as numerically superior bounds. Besides the improvements, this work aims to better popularize the aforementioned techniques within the CS community. Leveraging tools that have not been previously used in this context, particularly hypercontractive inequalities and results on concentration of sub-gamma distributions, we offer an elegant and modular analysis, as well as numerically superior bounds. Besides the improvements, this work aims to better popularize the aforementioned techniques within the CS community. Besides the improvements, this work aims to better popularize the aforementioned techniques within the CS community.