In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, to the form by using other well-defined sequences of integers obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients , and employing modular forms of higher levels. Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only recently that H. H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup , while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators. Levels 1–4A were given by Ramanujan (1914), level 5 by H. H. Chan and S. Cooper (2012), 6A by Chan, Tanigawa, Yang, and Zudilin, 6B by Sato (2002), 6C by H. Chan, S. Chan, and Z. Liu (2004), 6D by H. Chan and H. Verrill (2009), level 7 by S. Cooper (2012), part of level 8 by Almkvist and Guillera (2012), part of level 10 by Y. Yang, and the rest by H. H. Chan and S. Cooper. The notation jn(τ) is derived from Zagier and Tn refers to the relevant McKay–Thompson series. Level 1 Examples for levels 1–4 were given by Ramanujan in his 1917 paper. Given as in the rest of this article. Let, with the j-function j(τ), Eisenstein series E4, and Dedekind eta function η(τ). The first expansion is the McKay–Thompson series of class 1A () with a(0) = 744. Note that, as first noticed by J. McKay, the coefficient of the linear term of j(τ) almost equals 196883, which is the degree of the smallest nontrivial irreducible representation of the Monster group. Similar phenomena will be observed in the other levels. Define () Then the two modular functions and sequences are related by if the series converges and the sign chosen appropriately, though squaring both sides easily removes the ambiguity. Analogous relationships exist for the higher levels. Examples: where and is a fundamental unit. The first belongs to a family of formulas which were rigorously proven by the Chudnovsky brothers in 1989 and later used to calculate 10 trillion digits of π in 2011. The second formula, and the ones for higher levels, was established by H.H. Chan and S. Cooper in 2012. Level 2 Using Zagier's notation for the modular function of level 2, Note that the coefficient of the linear term of j2A(τ) is one more than 4371 which