A note on base change, identities involving \({\tau}(n)\), and a congruence of Ramanujan (Q1417928)

The theory of quadratic base change is used to derive new identities involving the Ramanujan function \(\tau(n)\) and the divisor sum \(\sigma_{11}(n)\). One of these determines \(\tau(17)\) directly from \(\tau(2)\), \(\tau(3)\), \(\tau(7)\), and \(\sigma_{11}(2)\). Another implies the Ramanujan congruence \(\tau(n)\equiv \sigma_{11}(n)\pmod{691}\).