Hecke eigenvalue of a nonholomorphic Eisenstein series

For the normalization

(T_{p} f) (τ) = p^{- 1} (f (p τ) + \sum_{b mod p} f ((τ + b) / p))

, one has

T_{p} G (τ, s) = (p^{s - 1} + p^{- s}) G (τ, s) .

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 4 / c / Solution

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