Modular forms with small Fourier coefficients

Computing the class number is a hard question. In 1956, Iwasawa announced a surprising formula for an infinite family of class numbers, starting an entire theory that lies behind this phenomenon. We will not focus too much on this theory (Iwasawa theory), but rather describe some analogous formulas for modular forms. Their origins have not been explained yet, especially when the $p$-th Fourier coefficient is small.