Eisenstein series of weight two ( $E_{2}$ )

E_{2} (τ) = 1 - 24 \sum_{n \geq 1} σ_{1} (n) q^{n}

is quasimodular:

E_{2} (- 1/ τ) = τ^{2} E_{2} (τ) + 6 τ / (πi)

.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 1 / b / Solution

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