Deriving identities involving divisor function by equating coefficients in appropriate identities in modular forms

Question

I am trying exercise problems of Apostol modular functions and Dirichlet series in number theory and I am struck on this problem of Ch -6 . I am adding its image.

I think ratio of modular discriminant should be taken , but modular discriminant is a cusp form so constant would be zero ( if I divide by modular form of appropriate weight) .

Can someone please help.

Answer 1

The identity arise since $E_{12}-E_6^2=k\Delta$, for a constant $k$ you should identify.

Then you have $$756\tau(n)=65\sigma_{11}(n)+691\sigma_5(n)-691\times252\sum(\text{omitted}).$$ Treating this modulo $691$ gives $$65\tau(n)\equiv65\sigma_{11}(n)\pmod{691}.$$ As $\gcd(65,691)=1$ then $$\tau(n)\equiv\sigma_{11}(n)\pmod{691}.$$