Elliptic integrals and Ramanujan-type series for 1/π associated with Γ0(N), where N is a product of two small primes

In the lost notebook Ramanujan stated results for elliptic integrals associated with Γ0(N) for N=5, 7, 10, 14, 15 and 35. But he did not record an integral corresponding to N=21 despite having worked out all of the relevant P–Q modular identities. Possibly, the reason for the omission is that the corresponding differentiation formula is more complicated. In this work, it is shown that under an appropriate change of variables, a particularly simple differentiation formula for N=21 can be obtained. It turns out to be just one piece of a larger theory, and we show how the level 21 theory can be extended to produce Ramanujan-type series for 1/π. We also outline how a similar procedure can be used to produce the corresponding results for N=22, 33 and 35.