Fourier coefficients and cuspidality of modular forms: a new approach

We provide a simple and new induction based treatment of the problem of distinguishing cusp forms from the growth of the Fourier coefficients of modular forms. Our approach gives the best possible ranges of the weights for this problem, and has wide adaptability. We propose a conjecture which asks the same converse question based on information on the Fourier-Jacobi coefficients, and answer it partially. We also discuss how to recover cuspidality from the poles of the allied Rankin-Selberg $L$-series.